The linear- and quadratic-in-spin contributions to the binding potential and gravitational-wave flux from binary systems are derived to next-to-next-to-leading order in the post-Newtonian (PN) expansion of general relativity, including finite-size and tail effects. The calculation is carried out through the worldline effective field theory framework. We find agreement in the overlap with the available PN and self-force literature. As a direct application, we complete the knowledge of spin effects in the evolution of the orbital phase for aligned-spin circular orbits to fourth PN order. We estimate the impact in the number of accumulated gravitational-wave cycles and find they make a significant contribution for next-generation observatories. The results presented here will therefore play an important role in providing reliable physical interpretation of gravitational-wave signals from spinning binaries with future gravitational-wave detectors such as LISA and the Einstein Telescope.

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• Effective field theory
• General relativity
• Gravitational waves
• Classical black holes