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  • Featured in Physics
  • Perspective
  • Open Access

Securing Elliptic Curve Cryptocurrencies against Quantum Vulnerabilities: Resource Estimates and Mitigations

Ryan Babbush1,*, Adam Zalcman1,†, Craig Gidney1,‡, Michael Broughton1, Tanuj Khattar1, Hartmut Neven1, Thiago Bergamaschi1,2, Justin Drake3, and Dan Boneh4

  • *Contact author: babbush@google.com
  • †Contact author: viathor@google.com
  • ‡Contact author: craiggidney@google.com
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PRX Quantum 7, 031001 – Published 21 August, 2026

DOI: https://doi.org/10.1103/j3xf-bw18

Abstract

The expected emergence of cryptographically relevant quantum computers (CRQCs) will represent a singular discontinuity in the history of digital security, with wide ranging impacts. This Perspective seeks to elucidate specific implications that the capabilities of developing quantum architectures have on blockchain vulnerabilities and potential mitigation strategies. First, we provide new resource estimates for breaking the 256-bit Elliptic Curve Discrete Logarithm Problem over the secp256k1 curve, the core of modern blockchain cryptography. We demonstrate that Shor’s algorithm for this problem can execute with either ≤1200 logical qubits and ≤90  million Toffoli gates or ≤1450 logical qubits and ≤70  million Toffoli gates. In the interest of responsible disclosure, we use a zero-knowledge proof to validate these results without disclosing attack vectors. On superconducting architectures with 10−3 physical error rates and planar connectivity, those circuits can execute in minutes using fewer than half a million physical qubits. We introduce a critical distinction between “fast-clock” (such as superconducting and photonic) and “slow-clock” (such as neutral atom and ion trap) architectures. Our analysis reveals that the first fast-clock CRQCs would enable “on-spend” attacks on public mempool transactions of some cryptocurrencies. We survey major cryptocurrency vulnerabilities through this lens, identifying systemic risks associated with advanced features in some blockchains such as smart contracts, proof-of-stake consensus, and data availability sampling mechanism, as well as the enduring concern of “abandoned” assets. We argue that technical solutions would benefit from accompanying public policy and discuss various frameworks of “digital salvage” to regulate the recovery or destruction of dormant assets while preventing adversarial seizure. We also discuss implications for other digital assets and tokenization as well as challenges and successful examples of the ongoing transition to post-quantum cryptography (PQC). Finally, we urge all vulnerable cryptocurrency communities to join the migration to PQC without delay.

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Physics Subject Headings (PhySH)

Focus

Quantum Hacking Coming Sooner than Expected

Published 21 August, 2026

New estimates suggest that some crypto-security systems may soon become vulnerable to quantum-powered break-ins.

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Popular Summary

Future quantum computers threaten to break the digital locks that secure modern cryptocurrencies, putting trillions of dollars at risk. By optimizing quantum algorithms, we drastically reduced the estimated computing power required to pick these digital locks, and we verified our results using a cryptographic proof that keeps the specific attack blueprints secret. Our calculations show that fast-acting quantum computers could figure out a user’s secret key in just a few minutes. This incredible speed would allow thieves to intercept and hijack active transfers before the network processes them, while also exposing massive pools of dormant accounts to theft. To avoid a catastrophic collapse of trust in the global digital economy, we urge the cryptocurrency industry and policymakers to urgently transition to quantum-resistant security standards.

Article Text

References (347)

  1. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  2. P. Schleich, L. M. Calderón, C. Sun, M. Bagherimehrab, A. Aldossary, J. S. Kottmann, and A. Aspuru-Guzik, Quantum Computing for Quantum Chemistry (American Chemical Society, Washington, DC, USA, 2025).
  3. R. Babbush, R. King, S. Boixo, W. Huggins, T. Khattar, G. H. Low, J. R. McClean, T. O’Brien, and N. Rubin, The grand challenge of quantum applications, PRX Quantum 7, 020101 (2025).
  4. A. Montanaro, Quantum algorithms: an overview, npj Quantum Inf. 2, 15023 (2016).
  5. J. Chou, J. Manyika, and H. Neven, The race to lead the quantum future, Foreign Affairs 104, 154 (2025), https://www.foreignaffairs.com/united-states/race-lead-quantum-future-chou-manyika-neven.
  6. D. J. Bernstein and T. Lange, Post-quantum cryptography, Nature (London) 549, 188 (2017).
  7. D. Joseph, R. Misoczki, M. Manzano, J. Tricot, F. D. Pinuaga, O. Lacombe, S. Leichenauer, J. Hidary, P. Venables, and R. Hansen, Transitioning organizations to post-quantum cryptography, Nature (London) 605, 237 (2022).
  8. C. Gidney, How to factor 2048 bit RSA integers with less than a million noisy qubits, arXiv:2505.15917.
  9. R. C. Merkle, Secure communications over insecure channels, Commun. ACM 21, 294 (1978).
  10. R. L. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM 21, 120 (1978).
  11. V. S. Miller, Use of elliptic curves in cryptography, in Advances in Cryptology—CRYPTO ’85 Proceedings, edited by H. C. Williams (Springer, Berlin, Heidelberg, 1986), pp. 417–426.
  12. N. Koblitz, Elliptic curve cryptosystems, Math. Comput. 48, 203 (1987).
  13. P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26, 1484 (1997).
  14. AMD-Google-Meta-Microsoft-NVIDIA, OCP GPU FW update specification, Technical Report, Open Compute Project, 2023.
  15. K. Schneider, L. Auer, and A. Wagner, Fault attacks on ECC signature verification, IACR Trans. Cryptographic Hardware Embedded Syst. 2025, 1010 (2025).
  16. E. Rescorla, The transport layer security (TLS) Protocol Version 1.3, Report No. RFC 8446, Internet Engineering Task Force, 2018, 10.17487/RFC8446.
  17. M. Marlinspike and T. Perrin, The X3DH key agreement protocol, Signal Technology Foundation, 2016, https://signal.org/docs/specifications/x3dh/.
  18. J. W. Bos, J. A. Halderman, N. Heninger, J. Moore, M. Naehrig, and E. Wustrow, Elliptic curve cryptography in practice, Financial Cryptography and Data Security, Lecture Notes in Computer Science 8437 (Springer, Berlin, 2013), 157, ISBN [Amazon][WorldCat], 10.1007/978-3-662-45472-5_11.
  19. YubiKey Technical Manual (Yubico, Stockholm, Sweden, 2026), https://docs.yubico.com/hardware/yubikey/yk-tech-manual/webdocs.pdf.
  20. D. Stebila and J. Green, Elliptic curve algorithm integration in the secure shell (SSH) transport layer, , 2009, 10.17487/RFC5656.
  21. P. Hoffman and W. Wijngaards, Elliptic curve digital signature algorithm (DSA) for DNSSEC, RFC 6605, , 2012, 10.17487/RFC6605, https://www.rfc-editor.org/info/rfc6605.
  22. N. Gura, A. Patel, A. Wander, H. Eberle, and S. C. Shantz, Comparing elliptic curve cryptography and RSA on 8-bit CPUs, in Cryptographic Hardware and Embedded Systems—CHES 2004, edited by M. Joye and J.-J. Quisquater (Springer, Berlin, Heidelberg, 2004), pp. 119–132.
  23. C. A. Lara-Nino, A. Diaz-Perez, and M. Morales-Sandoval, Elliptic curve lightweight cryptography: A survey, IEEE Access 6, 72514 (2018).
  24. E. Barker and S. P. N.I.S.T., NIST SP 800-57 Part 1 Rev. 5: Recommendation for Key Management: Part 1–General, NIST Special Publication 800-57 Part 1 Rev. 5 (National Institute of Standards and Technology, Gaithersburg, MD, USA, 2020).
  25. A. Milton and C. Shikhelman, Bitcoin and quantum computing: Current status and future directions, Research Report, Chaincode Labs, New York, NY, 2025.
  26. A. Li et al., The quantum threat to bitcoin, Human Rights Research Report, Human Rights Foundation, 2025, informed by technical insights from the 2025 Presidio Bitcoin Quantum Summit.
  27. C. Deegan, Quantum vulnerability of bitcoin addresses (2025), https://blog.projecteleven.com/posts/quantum-vulnerability-of-bitcoin-addresses.
  28. J. J. Pont, J. J. Kearney, J. Moyler, and C. A. Perez-Delgado, Downtime required for bitcoin quantum-safety, arXiv:2410.16965.
  29. D. Aggarwal, G. Brennen, T. Lee, M. Santha, and M. Tomamichel, Quantum attacks on bitcoin, and how to protect against them, Ledger 3, 68 (2018).
  30. T. Ruffing, The post-quantum security of bitcoin’s taproot as a commitment scheme, Cryptology (2025), Paper 2025/1307.
  31. S. Holmes and L. Chen, Assessment of quantum threat to bitcoin and derived cryptocurrencies, Cryptology (2021), Paper 2021/967.
  32. K. Fukuda, S. Matsuo, Y. Suga, and T. Ito, The grand challenge of PQC migration: Analysis of modern blockchain and intertwined human egoisms, Cryptology (2025), Paper 2025/1626.
  33. D. B. C. Costa, Post-quantum financial infrastructure framework (PQFIF): A roadmap for the quantum-safe transition of global financial infrastructure, regulatory submission (U.S. Securities and exchange commission (SEC), 2025) prepared for the U.S. Crypto assets task force–SEC.
  34. A. Maurushat, Disclosure of Security Vulnerabilities: Legal and Ethical Issues (Springer, London, 2013).
  35. ISO/I. E. C. I.S.O./I.E.C., 29147: 2018 Information Technology—Security Techniques Vulnerability Disclosure, Standard (International Electrotechnical Commission, Geneva, Switzerland, 2018).
  36. D. C. North, Institutions, J. Econ. Perspect. 5, 97 (1991).
  37. D. C. North, Institutions, Institutional Change and Economic Performance, Political Economy of Institutions and Decisions (Cambridge University Press, Cambridge, England, 1990).
  38. CoinMarketCap, FUD, CoinMarketCap Glossary (2021).
  39. BTSE. Crypto trading psychology: dealing with FUD and FOMO in the cryptocurrency market, Special Report, BTSE, 2023, BTSE Trading Psychology Series.
  40. R. Harris and A. McIntyre, The Complete Sales Letter Book: Model Letters for Every Selling Situation (Sharpe Professional, 1998).
  41. Project Eleven, Project eleven awards 1 BTC Q-day prize for largest quantum attack on elliptic curve cryptography to date (2026), https://blog.projecteleven.com/posts/project-eleven-awards-1-btc-q-day-prize-for-largest-quantum-attack-on-elliptic-curvecryptography- to-date.
  42. C. Gidney, The predictable failure of the QDay prize—Algassert (2026).
  43. S. Goldwasser, S. Micali, and C. Rackoff, The knowledge complexity of interactive proof-systems, in Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing, STOC’8 (Association for Computing Machinery, New York, NY, USA, 1985), pp. 291–304.
  44. J.-J. Quisquater, M. Quisquater, M. Quisquater, M. Quisquater, L. C. Guillou, M. A. Guillou, G. Guillou, A.-C. Guillou, G. Guillou, S. Guillou, and T. A. Berson, How to explain zero-knowledge protocols to your children, in Advances in Cryptology—CRYPTO ’89 Proceedings, Lecture Notes in Computer Science 435, edited by G. Brassard (Springer, New York, NY, 1990), ISBN [Amazon][WorldCat], 628, 10.1007/0-387-34805-0_60.
  45. S. Cu’ellar Gempeler, B. Harris, J. Parker, S. Pernsteiner, I. Sweet, and E. Tromer, Cheesecloth: Zero-knowledge proofs of real-world vulnerabilities, ACM Trans. Priv. Secur. 28, 1 (2025).
  46. D. Litinski, How to compute a 256-bit elliptic curve private key with only 50 million toffoli gates, arXiv:2306.08585.
  47. C. Gidney and M. Ekerå, How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits, Quantum 5, 433 (2021).
  48. M. Habov stiak, Hashed keys are actually fully quantum secure, bitcoin development mailing list (Google groups), 2025, mailing list thread featuring contributions from Lloyd Fournier, Agustin Cruz, Antoine Poinsot, and others. Proposes a commit-reveal scheme to protect hashed pubkeys from mempool-snatching by quantum adversaries.
  49. D. J. Bernstein, Introduction to post-quantum cryptography, Post-Quantum Cryptography, edited by D. J. Bernstein, J. Buchmann, and E. Dahmen (Springer, Berlin, Heidelberg, 2009), pp. 1–14.
  50. D. Bernstein and T. Lange, Post-quantum cryptography: Dealing with the fallout of physics success, Cryptology (International Association for Cryptologic Research, 2017).
  51. National Institute of Standards and Technology, Glossary: defense-in-depth (CSRC Computer Security Resource Center, 2026), https://csrc.nist.gov/glossary/term/defense_in_depth.
  52. P. Gretz, D. Gross, E. M. Haynes, Y. Tarek, A. Toma, A. Chitgupi, and T. Kurowski, Tokenization standards: taming the regulatory menagerie, Industry Research Report, Nethermind and PwC Germany, 2025, distributed in collaboration with Global Finance & Technology Network (GFTN).
  53. A. Yakovenko, Solana: A new architecture for a high performance blockchain, protocol perspective v0.8.13, Solana Labs, 2017.
  54. J. Chen and S. Micali, Algorand, arXiv:1607.01341.
  55. M. Lokhava, G. Losa, D. Mazières, G. Hoare, N. Barry, E. Gafni, J. Jove, R. Malinowsky, and J. McCaleb, Fast and secure global payments with stellar, in Proceedings of the 27th ACM Symposium on Operating Systems Principles, SOSP’19 (Association for Computing Machinery, New York, NY, USA, 2019), pp. 80–96.
  56. D. Mazières, The stellar consensus protocol: A federated model for internet-level consensus, Technical Report, Stellar Development Foundation, 2015.
  57. B. Chase and E. MacBrough, Analysis of the XRP ledger consensus protocol, arXiv:1802.07242.
  58. Ripple Developers, rippled: XRP ledger network server, GitHub Repository (2026).
  59. Algorand Foundation, Asset Tokenization on Algorand: Powering a more open Financial System (2026), https://algorand.co/solutions/tokenization.
  60. Stellar Development Foundation, RWA tokenization (2026), https://stellar.org/use-cases/tokenization.
  61. Team RippleX. The future of asset tokenization: A new token standard for institutional-grade finance on XRP ledger, Industry Research Report, Ripple Labs, Inc., San Francisco, CA, 2025. https://xrpl.org/static/pdf/Whitepaper_the_future_of_asset_tokenization.pdf
  62. D. O’Brien, Protecting chrome traffic with hybrid kyber KEM, chromium blog, 2023 (2026).
  63. L. Young, Technical brief: Quantum-resistant transactions on Algorand with Falcon signatures, Algorand developer blog (2025), technical contributions by Giulio Pizzini, Cosimo Bassi, and Steve Ferrigno.
  64. The QRL Foundation, Quantum resistant ledger: A visionary, future-proof blockchain with unparalleled security (2026).
  65. P. Waterland, Quantum Resistant Ledger (QRL), Technical White Paper, The QRL Foundation, Zug, Switzerland, 2016, https://github.com/theQRL/Whitepaper/blob/master/QRL_whitepaper.pdf.
  66. M. Zweil, Mochimo: Post-quantum currency, Protocol White Paper, Adequate Systems, LLC, Tampa, FL, 2018, https://mochimo.org/assets/files/mochimo_wp_EN.pdf.
  67. Alice, Bob, Eve, and Lambda, Abelian (ABEL)—A quantum-resistant cryptocurrency balancing privacy and accountability, Cryptographic White Paper, The Abelian Foundation, Singapore, 2022, https://download.pqabelian.io/release/docs/whitepaper.pdf.
  68. J. Dreyzehner, Quantumroot: quantum-secure vaults for bitcoin cash (2025), https://blog.bitjson.com/quantumroot/.
  69. ForkLog, ForkLog. XRP Ledger Implements Quantum Threat Protection (ForkLog Magazine, 2025).
  70. C. Deegan, A look at post quantum proposals for bitcoin 2025), https://blog.projecteleven.com/posts/a-look-at-post-quantum-proposals-for-bitcoin.
  71. F. Wang, S. Cohney, and J. Bonneau, SoK: Trusted setups for powers-of-tau strings, Cryptology (2025), Paper 2025/064.
  72. A. Pertsev, R. Semenov, and R. Storm, Tornado cash privacy solution, technical whitepaper version 1.4 (Tornado Cash, 2019).
  73. R. Acharya et al., Quantum error correction below the surface code threshold, Nature (London) 638, 920 (2024).
  74. M. L. Chan, A. A. Capatos, P. Lodahl, A. S. Sørensen, and S. Paesani, Practical blueprint for low-depth photonic quantum computing with quantum dots, arXiv:2507.16152.
  75. J. R. Scott and K. C. Balram, Timing constraints due to real-time graph-traversal algorithms on incomplete cluster states in photonic measurement-based quantum computing, Phys. Rev. Appl. 20 (2023).
  76. P. Stano and D. Loss, Review of performance metrics of spin qubits in gated semiconducting nanostructures, Nat. Rev. Phys. 4, 672 (2022).
  77. D. Bluvstein et al., A fault-tolerant neutral-atom architecture for universal quantum computation, Nature (London) 649, 39 (2025).
  78. V. M. Schäfer, C. J. Ballance, K. Thirumalai, L. J. Stephenson, T. G. Ballance, A. M. Steane, and D. M. Lucas, Fast quantum logic gates with trapped-ion qubits, Nature (London) 555, 75 (2018).
  79. S. Aaronson, More on whether useful quantum computing is “imminent”: Shtetl-optimized (2025), https://scottaaronson.blog/?p=9425.
  80. S. Aaronson, On reducing the cost of breaking RSA-2048 to 100,000 physical qubits (2026), https://scottaaronson.blog/?p=9564.
  81. A. Kerckhoffs, La cryptographie militaire, J. Des Sci. Militaires IX, 5 (1883).
  82. CERT Coordination Center, CERT/CC vulnerability disclosure policy, Official Policy Document (2026).
  83. Google Project Zero, Vulnerability disclosure policy (2026), https://projectzero.google/vulnerability-disclosure-policy.html.
  84. N. Carter, Bitcoin developers are mostly not concerned about quantum risk (2026), https://murmurationstwo.substack.com/p/bitcoin-developers-are-mostly-not.
  85. Succinct Labs, SP1: The Zero-Knowledge Virtual Machine (zkVM), (2026), https://github.com/succinctlabs/sp1.
  86. J. Jancar and V. Sedlacek, Parameters for secp256k1: Standard curves database, Technical specifications for the Koblitz curve used by Bitcoin and Ethereum (2026), https://std.neuromancer.sk/secg/secp256k1.
  87. D. R. L. Brown, SEC. 2: Recommended elliptic curve domain parameters, technical specification version 2.0 (Certicom research, 2010) standards for efficient cryptography (SEC).
  88. J. Proos and C. Zalka, Shor’s discrete logarithm quantum algorithm for elliptic curves (2003), 10.48550/ARXIV.QUANT-PH/0301141.
  89. C. Chevignard, P.-A. Fouque, and A. Schrottenloher, Reducing the number of qubits in quantum discrete logarithms on elliptic curves, Paper 2026/280, Cryptology (2026), https://eprint.iacr.org/2026/280.
  90. T. Häner, S. Jaques, M. Naehrig, M. Roetteler, and M. Soeken, Improved Quantum Circuits for Elliptic Curve Discrete Logarithms, Cryptology (2020), Paper 2020/077.
  91. Élie Gouzien, D. Ruiz, F.-M. Le Régent, Jérémie Guillaud, and N. Sangouard, Performance analysis of a repetition cat code architecture: Computing 256-bit elliptic curve logarithm in 9 hours with 126,133 cat qubits, Phys. Rev. Lett. 131 (2023).
  92. P.-L. Dallaire-Demers, W. Doyle, and T. Foo, Brace for impact: ECDLP challenges for quantum cryptanalysis (2025), arXiv:2508.14011.
  93. A. Fiat and A. Shamir, How to prove yourself: Practical solutions to identification and signature problems, Advances in Cryptology—CRYPTO’ 86, edited by A. M. Odlyzko (Springer, Berlin, Heidelberg, 1987), pp. 186–194.
  94. R. Babbush, A. Zalcman, C. Gidney, M. Broughton, T. Khattar, H. Neven, T. Bergamaschi, J. Drake, and D. Boneh, Data for “securing elliptic curve cryptocurrencies against quantum vulnerabilities: Resource estimates and mitigations” (2026).
  95. C. Jones, Low-overhead constructions for the fault-tolerant toffoli gate, Phys. Rev. A 87, 022328 (2013).
  96. A. G. Fowler, Time-optimal quantum computation (2012), arXiv:1210.4626.
  97. P. L. Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Math. Comput. 48, 243 (1987).
  98. H. Kim, K. Jang, S. Wang, A. Baksi, G. Song, H. Seo, and A. Chattopadhyay, New quantum circuits for ECDLP: Breaking prime elliptic curve cryptography in minutes, Cryptology 2026), Paper 2026/106.
  99. M. Roetteler, M. Naehrig, K. M. Svore, and K. Lauter, Quantum resource estimates for computing elliptic curve discrete logarithms, arXiv:1706.06752.
  100. C. Gidney, M. Newman, P. Brooks, and C. Jones, Yoked surface codes, arXiv:2312.04522.
  101. C. Gidney and M. Newman, Yoked surface codes, in Talk at the 7th International Conference on Quantum Error Correction (New Haven, Connecticut, USA, 2025).
  102. C. Gidney and A. G. Fowler, Flexible layout of surface code computations using autoccz states, arXiv:1905.08916.
  103. M. Ekerå. On factoring integers, and computing discrete logarithms and orders, quantumly, Ph.D. thesis, KTH Royal Institute of Technology, Stockholm, Sweden, 2024.
  104. C. Gidney, N. Shutty, and C. Jones, Magic state cultivation: Growing T states as cheap as cnot gates, arXiv:2409.17595.
  105. R. Babbush, R. King, S. Boixo, W. Huggins, T. Khattar, G. H. Low, J. R. McClean, T. O’Brien, and N. C. Rubin, The grand challenge of quantum applications, arXiv:2511.09124.
  106. B. Schneier, New attack on AES (2011), https://www.schneier.com/blog/archives/2011/08/new_attack_on_a_1.html.
  107. M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Elucidating reaction mechanisms on quantum computers, Proc. Natl. Acad. Sci. U.S.A. 114, 7555 (2017).
  108. Z. Li, J. Li, N. S. Dattani, C. J. Umrigar, and G. K..-L. Chan, The electronic complexity of the ground-state of the femo cofactor of nitrogenase as relevant to quantum simulations, J. Chem. Phys. 150 (2019).
  109. N. C. Jones, R. Van Meter, A. G. Fowler, P. L. McMahon, J. Kim, T. D. Ladd, and Y. Yamamoto, Layered architecture for quantum computing, Phys. Rev. X 2, 031007 (2012).
  110. A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
  111. J. O’Gorman and E. T. Campbell, Quantum computation with realistic magic-state factories, Phys. Rev. A 95, 032338 (2017).
  112. V. Gheorghiu and M. Mosca, Benchmarking the quantum cryptanalysis of symmetric, public-key and hash-based cryptographic schemes, arXiv:1902.02332.
  113. J. Lee, D. W. Berry, C. Gidney, W. J. Huggins, J. R. McClean, N. Wiebe, and R. Babbush, Even more efficient quantum computations of chemistry through tensor hypercontraction, PRX Quantum 2, 030305 (2021).
  114. V. von Burg, G. H. Low, T. Häner, D. S. Steiger, M. Reiher, M. Roetteler, and M. Troyer, Quantum computing enhanced computational catalysis, Phys. Rev. Res. 3, 033055 (2021).
  115. D. Rocca, C. L. Cortes, Jérôme F. Gonthier, P. J. Ollitrault, R. M. Parrish, G.-L. Anselmetti, M. Degroote, N. Moll, R. Santagati, and M. Streif, Reducing the runtime of fault-tolerant quantum simulations in chemistry through symmetry-compressed double factorization, J. Chem. Theory Comput. 20, 4639 (2024).
  116. G. H. Low, R. King, D. W. Berry, Q. Han, A. E. DePrince III, A. F. White, R. Babbush, R. D. Somma, and N. C. Rubin, Fast quantum simulation of electronic structure by spectral amplification, Phys. Rev. X 15, 041016 (2025).
  117. S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-tolerant quantum memory, Nature (London) 627, 778 (2024).
  118. Q. Xu, J. P. Bonilla Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasić, M. D. Lukin, L. Jiang, and H. Zhou, Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays, Nat. Phys. 20, 1084 (2024).
  119. T. J. Yoder, E. Schoute, P. Rall, E. Pritchett, J. M. Gambetta, A. W. Cross, M. Carroll, and M. E. Beverland, Tour de gross: A modular quantum computer based on bivariate bicycle codes, arXiv:2506.03094.
  120. P. Webster, L. Berent, O. Chandra, E. T. Hockings, N. Baspin, F. Thomsen, S. C. Smith, and L. Z. Cohen, The pinnacle architecture: Reducing the cost of breaking rsa-2048 to 100 000 physical qubits using quantum ldpc codes, arXiv:2602.11457.
  121. J. Olle, R. Zen, M. Puviani, and F. Marquardt, Simultaneous discovery of quantum error correction codes and encoders with a noise-aware reinforcement learning agent, npj Quantum Inf. 10, 126 (2024).
  122. A. Y. He and Z.-W. Liu, Discovering highly efficient low-weight quantum error-correcting codes with reinforcement learning, arXiv:2502.14372.
  123. S. Yang and R. Calderbank, Spatially-coupled qldpc codes, Quantum 9, 1693 (2025).
  124. E. Heilman, Changes to BIP-360—pay to quantum resistant hash (P2QRH), delving bitcoin, 2025, post #29 identifies the two cases arising from consideration of CRQC clock speed.
  125. P. Anderson and M. L. Tushman, Technological discontinuities and dominant designs: A cyclical model of technological change, Adm. Sci. Q. 35, 604 (1990).
  126. G. E. Moore, Cramming more components onto integrated circuits, Electronics 38, 114 (1965).
  127. R. Acharya et al., Suppressing quantum errors by scaling a surface code logical qubit, Nature (London) 614, 676 (2023).
  128. D. Awschalom et al., Development of quantum interconnects (quics) for next-generation information technologies, PRX Quantum 2 (2021).
  129. P. Escofet, S. B. Rached, S. Rodrigo, C. G. Almudever, E. Alarc’on, and S. Abadal, Interconnect fabrics for multi-core quantum processors: A context analysis, in Proceedings of the 16th International Workshop on Network on Chip Architectures, NoCArc ’23 (Association for Computing Machinery, New York, NY, USA, 2023), pp. 34–39, 10.1145/3610396.3623267.
  130. C. A. Pattison, G. Baranes, J. P. B. Ataides, M. D. Lukin, and H. Zhou, Fast quantum interconnects via constant-rate entanglement distillation, arXiv:2408.15936.
  131. Project Eleven, The Q-day prize: Break ECC with quantum (2025).
  132. H. Beast, E. Heilman, and I. F. Duke, BIP 360: Pay-to-Merkle-Root (P2MR), BIP 360, Bitcoin Improvement Proposals, 2024, https://github.com/bitcoin/bips/blob/master/bip-0360.mediawiki.
  133. C. Hoskinson, Why We Are Building Cardano: A Subjective Approach, Foundational Whitepaper (IOHK, 2017).
  134. A. Antonopoulos, Mastering Bitcoin: Unlocking Digital Cryptocurrencies (O’Reilly Media, 2014).
  135. G. Walker, Learn me a bitcoin (2026), https://learnmeabitcoin.com.
  136. G. Walker, Script: A mini programming language—learn me a bitcoin (2025), https://learnmeabitcoin.com/technical/upgrades/taproot/.
  137. P. Todd, REWARD offered for hash collisions for SHA1, SHA256, RIPEMD160 and other, Bitcointalk Forum (2013).
  138. Bitcoin Core Developers, Segregated witness wallet development guide (2026), https://bitcoincore.org/en/segwit_wallet_dev/
  139. R. Linus, BitVM: Compute anything on bitcoin, Technical White Paper, 2023, https://bitvm.org/bitvm.pdf.
  140. M. Friedenbach, Why Im Against Taproot and Recommend Against Activation on Bitcoin, Freicoin Substack, 2021.
  141. P. Wuille, J. Nick, and A. Towns, BIP. 341: Taproot: SegWit version 1 spending rules, Bitcoin Improvement Proposals, Status: Deployed. Consensus Layer (Soft Fork). Requires BIP, 340 and BIP. 341. Licensed under BSD-3-Clause, 2020.
  142. P. Wuille, J. Nick, and A. Towns, BIP. 342: Validation of taproot scripts, Bitcoin Improvement Proposals, Status: Deployed. Consensus Layer (Soft Fork). Requires BIP, 340 and BIP. 341. Licensed under BSD-3-Clause, 2020.
  143. A. Day and C. Bookman, Bitcoin in BigQuery: blockchain analytics on public data, google cloud blog, developed in collaboration with Kaggle (2018).
  144. S. Nakamoto, Bitcoin: A peer-to-peer electronic cash system (2008).
  145. CoinPhoton, Bitcoin network clears congestion after bullish rally, Binance Square, 2024, documents network congestion between december 2023 and january 2024 with a peak of over 194,000 unconfirmed transactions (2024).
  146. G. Walker, Block reward: Incentive for miners to mine blocks—learn me a bitcoin (2025), https://learnmeabitcoin.com/technical/upgrades/taproot/.
  147. P. Daian, S. Goldfeder, T. Kell, Y. Li, X. Zhao, I. Bentov, L. Breidenbach, and A. Juels, Flash boys 2.0: Frontrunning in decentralized exchanges, miner extractable value, and consensus instability, in 2020 IEEE Symposium on Security and Privacy (SP) (2020), pp. 910–927.
  148. A. Chon’e and M. D. I. Canals, Maximal extractable value: Implications for crypto markets, Regulatory Risk Analysis ESMA50-481369926-29744, European Securities and Markets Authority, 2025.
  149. S. Nichols, Marathon launches slipstream tech stack to process non-standard bitcoin transactions, Bitcoin Magazine (via Nasdaq) (2024).
  150. Project, Bitcoin risq list, technical database, a comprehensive audit of bitcoin addresses with exposed public keys 2026.
  151. R. Stütz, N. Stifter, M. Dragaschnig, B. Haslhofer, and A. Judmayer, Reuse of public keys across utxo and account-based cryptocurrencies, arXiv:2601.19500.
  152. P. Wuille, BIP. 32: Hierarchical deterministic wallets, BIP 32, Bitcoin Improvement Proposals, 2012, https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki.
  153. O. Sattath, On the insecurity of quantum bitcoin mining, Int. J. Inf. Secur. 19, 291 (2020).
  154. R. R. Nerem and D. R. Gaur, Conditions for advantageous quantum bitcoin mining, Blockchain: Res. Appl. 4, 100141 (2023).
  155. N. Vescovo, The Coming Quantum Threat to Bitcoin & Rootstock: Risks, Readiness, and Roadmap (Rootstock Labs Blog, 2025).
  156. E. Musk, @grok estimate the probability of quantum computing cracking SHA-256, X (formerly twitter) (2025).
  157. S. Neagle, How should bitcoiners view quantum computing? Bitcoin Magazine, Technical Analysis, 2025.
  158. R. Babbush, J. R. McClean, M. Newman, C. Gidney, S. Boixo, and H. Neven, Focus beyond quadratic speedups for error-corrected quantum advantage, PRX Quantum 2 (2021).
  159. C. Zalka, Grover’s quantum searching algorithm is optimal, Phys. Rev. A 60, 2746 (1999).
  160. Bitmain, Antminer S19 pro specifications, Bitmain Technologies Ltd.,2021.
  161. M. Amy, O. D. Matteo, V. Gheorghiu, M. Mosca, A. Parent, and J. Schanck, Estimating the cost of generic quantum pre-image attacks on SHA-2 and SHA-3, Cryptology (2016), paper 2016/992.
  162. D. I. Ilie, S. M. Werner, I. Stewart, and W. J. Knottenbelt, Unstable throughput: When the difficulty algorithm breaks, arXiv:2006.03044.
  163. D. Boneh, B. Lynn, and H. Shacham, Short signatures from the weil pairing, in Advances in Cryptology—ASIACRYPT 2001 (Springer, Berlin, Heidelberg, 2001), pp. 514–532.
  164. C. Reitwiessner, EIP-196: Precompiled contracts for addition and scalar multiplication on the elliptic curve altbn128, Ethereum Foundation, Ethereum improvement proposal 196, 2017.
  165. P. S. L. M. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, Cryptology (2005), paper 2005/133.
  166. J. Jancar and V. Sedlacek, Parameters for BLS12-381—standard curves database, technical specifications for the BLS12-381 pairing-friendly elliptic curve. (2026), https://std.neuromancer.sk/bls/BLS12-381.
  167. P. S. L. M. Barreto, B. Lynn, and M. Scott, Constructing elliptic curves with prescribed embedding degrees, Cryptology (2002), paper 2002/088.
  168. W. Diffie and M. Hellman, New directions in cryptography, IEEE Trans. Inf. Theory 22, 644 (1976).
  169. D.-E. Hopwood, S. Bowe, T. Hornby, and N. Wilcox, Zcash protocol specification, version 2025.6.3-84-gf58f5c [NU6], protocol specification, Version 2025.6.3-84-gf58f5c [NU6], Electric coin Co., Protocol Specification, 2026.
  170. K. M. Alonso and S. Noether koe. Zero to monero: Second edition—a technical guide to a private digital currency; for beginners, amateurs, and experts, 2nd ed. (Monero Research Lab, 2020), v2.0.0.
  171. D. Burkett, LIP. 004: One-sided transactions in MimbleWimble (Consensus layer), Litecoin project, litecoin improvement proposal 4, 2020.
  172. T. Elgamal, A public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inf. Theory 31, 469 (1985).
  173. T. P. Pedersen, Non-interactive and information-theoretic secure verifiable secret sharing, Advances in Cryptology—CRYPTO ’91, edited by J. Feigenbaum (Springer, Berlin, Heidelberg, 1992), pp. 129–140.
  174. T. Ruffing and G. Malavolta, Switch commitments: A safety switch for confidential transactions, Cryptology (2017), paper 2017/237.
  175. D. Pointcheval, Commitment schemes (TD. 16), course material, master parisien de recherche en informatique (MPRI), tutorial covering formal properties of Pedersen and ElGamal commitments (2017).
  176. A. Kate, G. M. Zaverucha, and I. Goldberg, Constant-size commitments to polynomials and their applications, Advances in Cryptology—ASIACRYPT 2010, edited by M. Abe (Springer, Berlin, Heidelberg, 2010), pp. 177–194.
  177. D. Ryan, D. Feist, F. D’Amato, H.-W. Wang, and A. Stokes, EIP-7594: PeerDAS—peer data availability sampling, Ethereum Foundation, Ethereum improvement proposal 7594,2024.
  178. J. Thaler, Proofs, arguments, and zero-knowledge, Found. Trends Privacy Secur. 4, 117 (2022).
  179. N. Bitansky, R. Canetti, A. Chiesa, and E. Tromer, From extractable collision resistance to succinct non-interactive arguments of knowledge, and back again, in Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, ITCS (Association for Computing Machinery, New York, NY, USA, 2012), Vol. 12, pp. 326–349.
  180. D. Koshelev, Generation of “Independent” Points on Elliptic Curves by Means of Mordell–weil Lattices, Cryptology (2022), Paper 2022/794.
  181. Wikipedia contributors, Nothing-up-my-sleeve number (2026).
  182. D. R. L. Brown, Rolling up sleeves when subversion’s in a field?, Cryptology (2020), Paper 2020/074.
  183. B. Bünz, J. Bootle, D. Boneh, A. Poelstra, P. Wuille, and G. Maxwell, Bulletproofs: Short proofs for confidential transactions and more, Cryptology (2017), Paper 2017/1066.
  184. L. Gudgeon, P. Moreno-Sanchez, S. Roos, P. McCorry, and A. Gervais, SoK: Layer-two blockchain protocols, Cryptology 2019), Paper 2019/360.
  185. Zcash Team, The pasta curves: Pallas and vesta, GitHub repository, technical specification and implementation of a cycle of prime-order elliptic curves designed for highly efficient recursive proof composition in halo 2 (2026).
  186. Contributors evm.codes, Precompiled contracts, interactive EVM Reference (Osaka hardfork spec), catalog of precompiled contracts on ethereum, including ECDSA recovery (0x01) and BLS12-381 operations (0x0b-0x11). Source code available at: https://github.com/duneanalytics/evm.codes (2026), https://github.com/duneanalytics/evm.codes.
  187. P. Saggese, M. Fröwis, S. Kitzler, B. Haslhofer, and R. Auer, Towards verifiability of total value locked (TVL) in decentralized finance, Bank for International Settlements, BIS Working Papers 1268.2025, https://www.bis.org/publ/work1268.pdf.
  188. J. Poon and T. Dryja, The bitcoin lightning network: Scalable off-chain instant payment, Lightning network, protocol Perspective DRAFT version: 0.5.9.2, 2016.
  189. V. Buterin, Ethereum: A next-generation smart contract & decentralized application platform, Ethereum Foundation, protocol Perspective, 2014.
  190. G. Wood, Ethereum: A Secure Decentralised Generalised Transaction Ledger, Formal Specification Version: efc5f9a (Ethereum Foundation, 2025).
  191. N. Szabo, Smart contracts: Building blocks for digital markets (2018).
  192. Y. Zhao. The cost of delay: Evidence from the ethereum transaction fee market Economics honors thesis, Duke University, Durham, NC, 2023.
  193. M. Kalinin, D. Ryan, and V. Buterin, EIP-3675: Upgrade consensus to proof-of-stake, Ethereum Foundation, ethereum improvement proposal 3675, 2021).
  194. V. Buterin, D. Hernandez, T. Kamphefner, K. Pham, Z. Qiao, D. Ryan, J. Sin, Y. Wang, and Y. X. Zhang, Combining ghost and casper, arXiv:2003.03052.
  195. A. Lewis-Pye and T. Roughgarden, Permissionless consensus, arXiv:2304.14701.
  196. D. Johnson, A. Menezes, and S. Vanstone, The elliptic curve digital signature algorithm (ECDSA), Int. J. Inf. Secur. 1, 36 (2001).
  197. S. Palladino, F. Giordano, and H. Croubois (ERC-1967: Proxy storage slots, Ethereum Foundation, Ethereum Improvement Proposal 1967, 2019.
  198. V. Buterin, Y. Weiss, D. Tirosh, S. Nacson, A. Forshtat, K. Gazso, and T. Hess, ERC-4337: Account abstraction using alt Mempool [DRAFT], Ethereum Foundation, Ethereum Improvement Proposal 4337, 2021.
  199. A. Day and E. Medvedev, Ethereum in BigQuery: A public dataset for smart contract analytics, Google Cloud Blog, developed in collaboration with Kaggle, AMPOS, and the ethereum ETL project (2018).
  200. V. Buterin, S. Wilson, A. Dietrichs, and lightclient, EIP-7702: Set code for EOAs, Ethereum Foundation, ethereum improvement proposal 7702, 2024.
  201. F. Vogelsteller and V. Buterin, ERC-20: Token standard, Ethereum Foundation, ethereum improvement proposal 20, 2015.
  202. W. Entriken, D. Shirley, J. Evans, and N. Sachs, ERC-721: Non-fungible token standard, Ethereum Foundation, ethereum improvement proposal 721, 2018.
  203. J. Lebrun, T. Malghem, K. Thizy, L. Falempin, and A. Boudjemaa, ERC-3643: T-REX—token for regulated EXchanges, Ethereum Foundation, Ethereum improvement proposal 3643, 2021.
  204. L. Fink and R. Goldstein, Larry Fink and Rob Goldstein on how tokenisation could transform finance, The Economist by Invitation (2025), https://www.blackrock.com/corporate/literature/article-reprint/larry-fink-rob-goldstein-economist-op-ed-tokenization.pdf.
  205. Castle Labs, Inc., Real-world asset analytics dashboard (RWA.xyz) (2026).
  206. A. Zamyatin, M. Al-Bassam, D. Zindros, E. Kokoris-Kogias, P. Moreno-Sanchez, A. Kiayias, and W. J. Knottenbelt, Sok: communication across distributed ledgers, in Financial Cryptography and Data Security: 25th International Conference, FC 2021, Virtual Event, March 1-5, 2021, Revised Selected Papers, Part II (Springer-Verlag, Berlin, Heidelberg, 2021), pp. 3–36.
  207. E. Ben-Sasson, I. Bentov, Y. Horesh, and M. Riabzev, Scalable zero knowledge with no trusted setup, in Advances in Cryptology–CRYPTO 2019: 39th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 18-22, 2019, Proceedings, Part III (Springer-Verlag, Berlin, Heidelberg, 2019), pp. 701–732.
  208. J. Kempton, EIP-7932: Secondary signature algorithms [DRAFT], Ethereum Foundation, Ethereum improvement proposal 7932, 2025.
  209. L2BEAT sp. z o.o., L2beat: The state of the layer two ecosystem (2026).
  210. V. Buterin and V. Griffith, Casper the friendly finality gadget, arXiv:1710.09437.
  211. F. D’Amato, B. Monnot, M. Neuder, J. Traglia Potuz, and T. Tsao, EIP-7732: Enshrined proposer-builder separation [DRAFT], Ethereum Foundation, Ethereum improvement proposal 7732, 2024.

  212. Hildobby, Ethereum ETH staking dashboard, Dune Analytics, industry-standard real-time tracker for Ethereum staking metrics (2026).

  213. J. Drake, D. Khovratovich, M. Kudinov, and B. Wagner, Hash-based multi-signatures for post-quantum ethereum, Cryptology (2025), Paper 2025/055.
  214. V. Buterin, D. Feist, D. Loerakker, G. Kadianakis, M. Garnett, M. Taiwo, and A. Dietrichs, EIP-4844: Shard Blob Transactions, Ethereum Foundation, Ethereum Improvement Proposal 4844, 2022.
  215. Ethereum Foundation, The KZG trusted setup ceremony, Official Protocol Website, a massively multi-party computation to produce the structured Reference string for KZG polynomial commitments used in EIP-4844 (2022).
  216. E. Ben-Sasson, I. Bentov, Y. Horesh, and M. Riabzev, Fast reed-solomon interactive oracle proofs of proximity, Schloss dagstuhl–Leibniz-zentrum für informatik, 2018.
  217. M. Hall-Andersen, M. Simkin, and B. Wagner, Foundations of data availability sampling, Cryptology (2023), paper 2023/1079.
  218. Ethereum Foundation, Official website of the ethereum Foundation (2026), https://ethereum.foundation/.
  219. A. Antonopoulos and G. D, Mastering ethereum: Building smart contracts and DApps, 1st ed. (O’Reilly Media, 2018).
  220. C. Detrio, EIP-779: Hardfork meta: DAO fork, Ethereum Foundation, Ethereum improvement proposal 779, 2017.
  221. U.S. Securities and Exchange Commission, Report of investigation pursuant to Section 21(a) of the securities exchange act of 1934: the DAO, U.S. Securities and Exchange Commission, Report of Investigation Release No. 81207, 2017.
  222. A. Schoedon, EIP-999: Restore contract code at 0x863DF6BFa4469f3ead0bE8f9F2AAE51c91A907b4 [DRAFT], Ethereum Foundation, Ethereum improvement proposal 999, 2018.
  223. J. Drake, Today marks an inflection in the Ethereum Foundation’s long-term quantum strategy, 2026, https://x.com/drakefjustin/status/2014791629408784816.
  224. L. Parker, Stepping back, GitHub Gist, 2024, technical manifesto calling for a moratorium on quantum-vulnerable R&D and the immediate development of a fallback post-quantum Monero protocol.
  225. SyntheticBird, SyntheticBird, Discussion: post-quantum security and ethical considerations over elliptic curve cryptography, monero research lab (MRL), GitHub issue #131, 2024, formalizes the proposal for a 5-year post-quantum migration timeline, citing ethical responsibilities and the google willow quantum processor announcement.
  226. J. Jefferson, DIP: OP_CHECKZKP zero-knowledge upgrade, dogecoin improvement proposal (Draft), GitHub issue #3869, proposes a new opcode for onchain verification of SNARKs to enable zk-rollups on dogecoin (2025).
  227. A. Back, M. Corallo, L. Dashjr, M. Friedenbach, G. Maxwell, A. Miller, A. Poelstra, J. Tim’on, and P. Wuille, Enabling blockchain innovations with pegged sidechains, protocol Perspective commit: 5620e43, Blockstream, 2014.
  228. A. Singh, K. Click, R. M. Parizi, Q. Zhang, A. Dehghantanha, and K.-K. R. Choo, Sidechain technologies in blockchain networks: An examination and state-of-the-art review, J. Netw. Comput. Appl. 149, 102471 (2020).
  229. S. D. Lerner, J. A. Cid-Fuentes, J. Len, R. Fernàndez-València, P. Gallardo, N. Vescovo, R. Laprida, S. Mishra, F. Jinich, and D. Masini, RSK: A Bitcoin Sidechain with Stateful Smart-Contracts, Cryptology (2022), Paper 2022/684.
  230. S. D. Lerner, RSK: bitcoin powered smart contracts, protocol Perspective revision 11, RSK labs, 2019.
  231. Tom Elvis Jedusor, MIMBLEWIMBLE, original text file distributed via IRC (2016).
  232. A. Yang, D. Burkett, and C. Lee, LIP. 003: MimbleWimble via extension blocks (Consensus layer), Litecoin Project, litecoin improvement proposal 3, 2019.
  233. T. Ruffing, Switch commitments (again), MimbleWimble mailing list archive, proposes to commit with C=vG+bH where b=b′+hash(vG+b′H,b′J), so C can be treated as either Pedersen or ElGamal commitment (2018).
  234. E. Ben-Sasson, A. Chiesa, E. Tromer, and M. Virza, Succinct non-interactive zero knowledge for a von Neumann architecture, Cryptology (2013), paper 2013/879.
  235. D.-E. Hopwood, ZIP. 211: Disabling addition of new value to the sprout chain value pool, Electric Coin Co., Zcash Improvement Proposal 211, 2019.
  236. Y. T. Lai and Zcash Developers, Pull request #4489: Disable addition of new value to the sprout value pool, Zcash GitHub Repository (2020), implements ZIP.
  237. J. Groth, On the size of pairing-based non-interactive arguments, Cryptology (2016), Paper 2016/260.
  238. A. Miller and S. Bowe, Announcing the world’s largest multi-party computation ceremony, Zcash Foundation Blog, 2017, official announcement of the powers of tau MPC ceremony for the sapling upgrade.
  239. Electric Coin Co., The halo2 book (2026).
  240. Y. Nir and A. Langley, ChaCha20 and Poly1305 for IETF protocols, RFC 7539, Internet Engineering Task Force, 2015, 10.17487/RFC7539.
  241. D.-E. Hopwood and J. Grigg, ZIP 2005: Quantum recoverability, Zcash Improvement Proposal (Draft), 2025, credits: Sean Bowe. Discussion available at GitHub issue #1135 (2005), https://github.com/zcash/zips/issues/1135.
  242. S. Bowe, Zcash and quantum computers (2025), https://seanbowe.com/blog/zcash-and-quantum-computers/.
  243. Electric Coin Company, Turnstile enforcement against counterfeiting, official zcash blog (2019).
  244. S. Bowe and D.-E. Hopwood, ZIP. 209: Prohibit negative shielded chain value pool balances (2019).
  245. D. A. Cooper, D. C. Apon, Q. H. Dang, M. S. Davidson, M. J. Dworkin, and C. A. Miller, Recommendation for stateful hash-based signature schemes (2020).
  246. L. Ducas, T. Lepoint, V. Lyubashevsky, P. Schwabe, G. Seiler, and D. Stehle, CRYSTALS–dilithium: Digital signatures from module lattices, Cryptology (2017), Paper 2017/633.
  247. National Institute of Standards and Technology, Module-Lattice-Based Digital Signature Standard, FIPS PUB 204, National Institute of Standards and Technology, Gaithersburg, MD, 2024, 10.6028/NIST.FIPS.204
  248. R. C. Merkle, A certified digital signature, in Advances in Cryptology—CRYPTO’ 89 Proceedings, edited by G. Brassard (Springer New York, New York, NY, 1990), pp. 218–238.
  249. C. Dods, N. P. Smart, and M. Stam, Hash based digital signature schemes, in Proceedings of the 10th International Conference on Cryptography and Coding, IMA’05 (Springer-Verlag, Berlin, Heidelberg, 2005), pp. 96–115.
  250. Algorand Foundation, Algorand TEAL: Transaction execution approval language, Algorand developer documentation (2026).
  251. Algorand Foundation, Algorand virtual machine, algorand developer documentation (2026).
  252. R. Perlner, FIPS. 206 Status Update, Technical Presentation, in 6th PQC Standardization Conference in Gaithersburg, Maryland (2025), https://csrc.nist.gov/csrc/media/presentations/2025/fips-206-fn-dsa-(falcon)/images-media/fips_206-perlner_2.1.pdf, 2025.
  253. P.-A. Fouque, J. Hoffstein, P. Kirchner, V. Lyubashevsky, T. Pornin, T. Prest, T. Ricosset, G. Seiler, W. Whyte, and Z. Zhang, Falcon: Fast-fourier lattice-based compact signatures over NTRU, technical specification (NIST PQC submission) specification v1.2, NIST Post-Quantum Cryptography Project, 2020.
  254. Algorand Foundation, Algorand TEAL opcodes: falcon_verify, lgorand eveloper Reference, 2026, bytecode 0x85, availabile in v12.
  255. Algorand Foundation, Rekeying Accounts (Algorand Developer Documentation, 2026).
  256. Bitcoin Stack Exchange, Answer to “Any innovations in altcoin tech for Bitcoin’s Script language?” (2022), https://Bitcoin.stackexchange.com/a/115856.
  257. CashScript Developers, Online documentation for the CashScript programming language, https://cashscript.org/docs/basics/about.
  258. CashTokens Developers, Online documentation for the CashTokens protocol (2026), https://cashtokens.org/docs/intro/.
  259. D. McGrew, M. Curcio, and S. Fluhrer, Leighton-micali hash-based signatures, RFC 8554, , 2019, 10.17487/RFC8554.
  260. Bitcoin Cash Research, Quantum-resistant bitcoin cash: A challenge-based transition protocol (2026).
  261. D. Little, Silur, and I. E. Ashimine, Solana Winternitz Vault: A quantum-resistant lamport vault using WOTS (GitHub Repository, 2026).
  262. M. Swayne, Solana takes a step toward PQC era with quantum-resistant vault, The Quantum Insider (2025).
  263. Castle Labs, Inc., OpenEden T-bill (TBILL) vault—real-world asset analytics dashboard (RWA.xyz) (2026), https://app.rwa.xyz/assets/TBILL, observed 61.8 million USD on the XRP Ledger out of total supply of 92.5 million USD.
  264. C. Goes, The interblockchain communication protocol: An overview, arXiv:2006.15918.
  265. Circle Internet Financial, Circle Is Discontinuing Support for USDC on the TRON Blockchain (Official Blog Announcement, 2024).
  266. The Q. R. L. Foundation, QRL. 2.0 (Codenamed project zond), official project roadmap (2026).
  267. M. M. T. Chakravarty, J. Chapman, K. MacKenzie, O. Melkonian, M. Peyton Jones, and P. Wadler, The extended utxo model, Financial Cryptography and Data Security, edited by M. Bernhard, A. Bracciali, L. J. Camp, S. Matsuo, A. Maurushat, P. B. Rønne, and M. Sala (Springer International Publishing, Cham, 2020), pp. 525–539.
  268. V. L. s.r.o., Understanding Cardano certificates, Cexplorer.io, 2023, technical analysis of Cardano’s stake pool and delegation certificates, detailing the separation of payment and staking credentials.
  269. TRON. D.A.O., Account Permission Management (TRON Developer Network, 2026).
  270. Ripple, Documentation—Best Practices—Key Management—Remove a Regular Key Pair, XRP Ledger Dev Portal (2026).
  271. G. Alagic, L. Chen, S. Jordan, Y.-K. Liu, D. Moody, R. Peralta, R. Perlner, and D. Sonneman, Transition to Post-quantum cryptography standards, NIST Internal Report (Initial Public Draft), NIST IR 8547, National Institute of Standards and Technology, 2024.
  272. R. J. McEliece, A public-key cryptosystem based on algebraic coding theory, Deep Space Network Prog. Rep. 44, 114 (1978).
  273. L. Lamport, Constructing digital signatures from a one way function, Tech. Rep. CSL-98, in Proceedings of HICSS-43 (1979).
  274. R. C. Merkle. Secrecy, authentication, and public key systems, Ph.D. thesis, Stanford University, Stanford, CA, USA, 1979.
  275. L. Chen, S. Jordan, Y.-K. Liu, D. Moody, R. Peralta, R. Perlner, and D. Smith-Tone, Report on post-quantum, Cryptography (2016).
  276. C. Peikert, A decade of lattice cryptography, Cryptology (2015), Paper 2015/939.
  277. E. Fathalla and M. Azab, Beyond classical cryptography: A systematic review of post-quantum hash-based signature schemes, security, and optimizations, IEEE Access 12, 175969 (2024).
  278. V. Weger, N. Gassner, and J. Rosenthal, A survey on code-based cryptography, arXiv:2201.07119.
  279. J. Ding, J. Gower, and D. Schmidt, Multivariate Public Key Cryptosystems (2006), Vol. 25.
  280. L. De Feo, Mathematics of isogeny based cryptography, arXiv:1711.04062.
  281. D. Jao and L. De Feo, Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies, in Post-Quantum Cryptography (Springer, Berlin, Heidelberg, 2011), pp. 19–34.
  282. W. Castryck and T. Decru, An efficient key recovery attack on SIDH, Cryptology (2022), paper 2022/975.
  283. O. Regev, On lattices, learning with errors, random linear codes, and cryptography, in Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’05(Association for Computing Machinery, New York, NY, USA, 2005), pp. 84–93.
  284. S. P. Jordan, N. Shutty, M. Wootters, A. Zalcman, A. Schmidhuber, R. King, S. V. Isakov, T. Khattar, and R. Babbush, Optimization by decoded quantum interferometry, Nature (London) 646, 831 (2025).
  285. M. Ajtai, Generating hard instances of the short basis problem, Automata, Languages and Programming, edited by J. Wiedermann, P. van Emde Boas, and M. Nielsen (Springer, Berlin, Heidelberg, 1999), pp. 1–9.
  286. A. Wein, A. El Alaoui, and C. Moore, The kikuchi hierarchy and tensor pca, J. ACM 72, 1 (2025).
  287. M. B. Hastings, Classical and quantum algorithms for tensor principal component analysis, Quantum 4, 237 (2020).
  288. A. Schmidhuber, R. O’Donnell, R. Kothari, and R. Babbush, Quartic quantum speedups for planted inference, Phys. Rev. X 15 (2025).
  289. National Institute of Standards and Technology, Module-lattice-based key-encapsulation mechanism standard, , Gaithersburg, MD, 2024, 10.6028/NIST.FIPS.203.
  290. National Institute of Standards and Technology, Stateless hash-based digital signature standard, , Gaithersburg, MD, 2024, 10.6028/NIST.FIPS.205.
  291. S. Marzougui, I. Kabin, J. Krämer, T. Aulbach, and J.-P. Seifert, On the feasibility of single-trace attacks on the gaussian sampler using a CDT, Cryptology (2023), paper 2023/142.
  292. PQShield. Post-Quantum Signatures Zoo, GitHub Pages (2026).
  293. L. D. Feo, D. Kohel, A. Leroux, C. Petit, and B. Wesolowski, SQISign: Compact post-quantum signatures from quaternions and isogenies, Cryptology (2020), Paper 2020/1240.
  294. H. Beast, P2QRH / BIP-360 update, bitcoin development mailing list (Google groups), discussion of BIP-360 featuring contributions from Matt Corallo, Jonas Nick, and others. Highlights the DDoS risk of Post-quantum transition scripts in bitcoin (2025).
  295. P. C. van Oorschot and M. J. Wiener, Parallel collision search with cryptanalytic applications, J. Cryptol. 12, 1 (1999).
  296. N. Bindel and B. Hale, A note on hybrid signature schemes, Cryptology (2023), Paper 2023/423.
  297. M. A. Aardal, D. F. Aranha, K. Boudgoust, S. Kolby, and A. Takahashi, Aggregating falcon signatures with LaBRADOR, Cryptology (2024), Paper 2024/311.
  298. Y. Quan, Improving bitcoin’s Post-quantum transaction efficiency with a novel lattice-based aggregate signature scheme based on CRYSTALS-dilithium and a STARK protocol, IEEE Access 10, 132472 (2022).
  299. N. Bitansky, R. Canetti, A. Chiesa, and E. Tromer, Recursive composition and bootstrapping for SNARKs and proof-carrying data, Cryptology (2012), Paper 2012/095.
  300. I. Stewart, D. Ilie, A. Zamyatin, S. Werner, M. F. Torshizi, and W. J. Knottenbelt, Committing to quantum resistance: A slow defence for bitcoin against a fast quantum computing attack, R. Soc. Open Sci. 5, 180410 (2018).
  301. N. Sklavos and O. Koufopavlou, Implementation of the sha-2 hash family standard using fpgas, J. Supercomput. 31, 227 (2005).
  302. S. D. Lerner, The well deserved fortune of Satoshi Nakamoto, bitcoin creator, visionary and genius, 2013, https://bitslog.com/2013/04/17/the-well-deserved-fortune-of-satoshi-nakamoto/.
  303. M. Humayun and R. Belk, “Satoshi is dead. Long live Satoshi”: The curious case of bitcoin’s creator (2018), pp. 19–35.
  304. K. Finlow-Bates, In search of lost keys, Medium (2023).
  305. P. McCormack and D. Held, Bitcoin’s immaculate conception: Audio interview transcription—WBD043, medium: HackerNoon, 2018, interview date: oct 25-discusses the philosophical significance of Satoshi Nakamoto’s disappearance and the “Immaculate conception” theory of bitcoin’s launch (2018).
  306. Officer’s Notes, New theory by BTCparser: Who might be the real Satoshi? answering the question of why he never touched his 2009 bitcoins!, Medium: coinmonks (2024).
  307. D. Bradbury, The problem with bitcoin, Comput. Fraud Secur. 2013, 5 (2013).
  308. J. Bartusek. Obfuscation of quantum computation, Ph.D. thesis, EECS Department, University of California, Berkeley, 2024.
  309. M.-Y. Huang and E.-C. Tang, Obfuscation of unitary quantum programs, arXiv:2507.11970.
  310. Luca Martini, Matthias Heuser, Martin Fest, Uwe Stegemann, Sebastian Schneider, Gabriel Brenna, Eckart Windhagen, and Thomas Poppensieker, Bad Banks: Finding the Right Exit from the Financial Crisis, Working Paper 12, Working Papers on Risk, New York, NY, 2009).
  311. H. Beast and M. Casey, BIP draft: Hourglass spending rulesb, 2025, https://github.com/cryptoquick/bips/blob/hourglass/bip-hourglass.mediawiki.
  312. M. Palatinus, P. Rusnak, A. Voisine, and S. Bowe, BIP. 39: Mnemonic code for generating deterministic keys,, BIP 39, , 2013, https://github.com/bitcoin/bips/blob/master/bip-0039.mediawiki.
  313. C. Deegan, Hello Yellowpages (Project Eleven Blog, 2025), https://blog.projecteleven.com/posts/hello-yellowpages.
  314. J. Nick, A. Poelstra, and G. Sanders, Liquid: A Bitcoin Sidechain, Protocol Perspective (Blockstream, 2020).
  315. E. Voskuil, Cryptoeconomics: Fundamental principles of bitcoin, 2nd ed. (Eric Voskuil, United States, 2020), version 1.2.3 (PDF), Edited and illustrated by James Chiang.
  316. M. Nadler and F. Schär, Tornado cash and blockchain privacy: A primer for economists and policymakers, Federal Reserve Bank of St. Louis Review (2023). first Quarter 2023.
  317. A. Wahrstätter, J. Ernstberger, A. Yaish, L. Zhou, K. Qin, T. Tsuchiya, S. Steinhorst, D. Svetinovic, N. Christin, M. Barczentewicz, and A. Gervais, Blockchain censorship, in Proceedings of the ACM Web Conference 2024, WWW ’24 (Association for Computing Machinery, New York, NY, USA, 2024), pp. 1632–1643.
  318. A. Brownworth, J. Durfee, M. J. Lee, and A. Martin, Regulating decentralized systems: Evidence from sanctions on tornado cash, Staff Reports 1112, Federal Reserve Bank of New York, 2024.
  319. N. Tovanich, N. Soulie, N. Heulot, and P. Isenberg, The evolution of mining pools and miners’ behaviors in the bitcoin blockchain, IEEE Trans. Networks Serv. Manage. 19, 3633 (2022).
  320. S. Shen and A. Galbraith, China’s Ban Forces Some Bitcoin Miners to Flee Overseas, Others Sell Out (Reuters, 2021).
  321. Reuters, US Becomes Largest Bitcoin Mining Centre following China Ban (Reuters, 2021).
  322. Cambridge Centre for Alternative Finance, Cambridge bitcoin electricity consumption index (CBECI): Mining map, interactive data dashboard, approximate geographic distribution of bitcoin hashrate and estimated total network electricity consumption (2026).
  323. G. Walker, Taproot: Technical explanation—learn me a bitcoin (2026).
  324. C. S. Gokhale and A. Traulsen, Evolutionary games in the multiverse, Proc. Natl. Acad. Sci. U.S.A. 107, 5500 (2010), https://www.pnas.org/doi/pdf/10.1073/pnas.0912214107.
  325. W. Poundstone, Prisoner’s Dilemma (Doubleday, 1992).
  326. G. H. Snyder, “prisoner’s dilemma” and “chicken” models in international politics, Int. Stud. Q. 15, 66 (1971), https://academic.oup.com/isq/article-pdf/15/1/66/5096260/15-1-66.pdf.
  327. R. O. Keohane, After hegemony: Cooperation and discord in the world political economy, revised ed. (Princeton University Press, 1984).
  328. W. Kenton, Escheat: Meaning, process, and reclaiming assets, Investopedia, 2025, reviewed by Toby Walters, defines “escheat” as “when a government obtains ownership of unclaimed property or estate assets due to there being no identifiable heirs or beneficiaries”.
  329. Merriam-Webster, Definition of laches, dictionary, defines “laches” as “undue delay in asserting a legal right or privilege”, Merriam-webster.com.
  330. , Revised uniform unclaimed property act, uniform law commission, last updated in 2016; originally promulgated in 1954 (2016).
  331. G. Walker, OP_RETURN: Storing data on the blockchain—Learn Me A Bitcoin (2025).
  332. L. Lessig, Code and Other Laws of Cyberspace (Basic Books, New York, NY, 1999).
  333. A. L. Russell, ’Rough consensus and running code’ and the internet-OSI standards war, IEEE Ann. Hist. Comput. 28, 48 (2006).
  334. K. Ryan, We beat Google’s zero-knowledge proof of quantum cryptanalysis (The Trail of Bits Blog, 2026).
  335. O. Goldreich and Y. Oren, Definitions and properties of zero-knowledge proof systems, J. Cryptol. 7, 1 (1994).
  336. C. Gidney, Verifying measurement based uncomputation—Algassert (2019).
  337. P. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, in Proceedings 35th Annual Symposium on Foundations of Computer Science (1994), pp. 124–134.
  338. M. Ekerå, Revisiting shor’s quantum algorithm for computing general discrete logarithms, arXiv:1905.09084 (2019).
  339. R. B. Griffiths and C.-S. Niu, Semiclassical fourier transform for quantum computation, Phys. Rev. Lett. 76, 3228 (1996).
  340. D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, Qubitization of arbitrary basis quantum chemistry leveraging sparsity and low rank factorization, Quantum 3, 208 (2019).
  341. C. Gidney, Halving the cost of quantum addition, Quantum 2, 74 (2018).
  342. C. Zalka, Shor’s algorithm with fewer (pure) qubits (2006).
  343. National Institute of Standards and Technology, SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions, , Gaithersburg, MD, 2015, 10.6028/nist.fips.202.
  344. E. Ben-Sasson, I. Bentov, Y. Horesh, and M. Riabzev, Scalable, transparent, and post-quantum secure computational integrity, Cryptology (2018), Paper 2018/046.
  345. P. Valiant, Incrementally verifiable computation or proofs of knowledge imply time/space efficiency, Theory of Cryptography, edited by R. Canetti (Springer, Berlin, Heidelberg, 2008), pp. 1–18.
  346. L. Goldberg, S. Papini, and M. Riabzev, Cairo–a turing-complete STARK-friendly CPU architecture, Cryptology (2021), Paper 2021/1063.
  347. Succinct Labs, (Groth16, PLONK, and the zero-knowledgeness of SP1, official documentation, clarifies the zero-knowledge properties of SP1 proofs, noting that Groth16/PLONK proofs are ZK and STARK proofs currently are not (2026).

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PRX Quantum 7, 031001– Published 21 August, 2026
  • Received 8 April 2026
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