The fractional Josephson effect (a $4\pi$-periodic current-phase relation, CPR) serves as a definitive signature of topological superconductivity mediated by chiral Majorana modes (CMMs). Here, we propose a cascaded topological heterostructure comprising two stacked proximity-induced topological superconductor (TSC)/quantum anomalous Hall insulator/TSC Josephson junctions under a transverse electric field. A rigid $4\pi$-periodic supercurrent emerges due to CMM interference, exhibiting two distinct transport signatures: (1) an intrinsic 0-$\pi$ state transition governed by the central superconductor’s phase; (ii) an extrinsic anomalous Josephson effect modulated by the transverse electric field. We demonstrate that the $4\pi$-periodic CPR synchronizes with zero-energy crossing displacements of nonlocal Andreev bound states, arising from nonlocal Andreev reflections mediated by CMMs. The study provides experimental designs to verify CMMs via both the fractional Josephson effect and quantized crossed Andreev reflection conductance, with applications in phase-controllable Josephson devices and topological transistors.

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• Majorana bound states
• Quantum anomalous Hall effect
• Quantum transport
• Josephson junctions
• Superconducting devices
• Topological insulators
• Topological superconductors
• Chiral symmetry