  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 literature and test-body limit. 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 of the new results in the number of accumulated
gravitational-wave cycles. We find they will play an important role in
providing reliable physical interpretation of gravitational-wave signals from
spinning binaries with future detectors such as LISA and the Einstein
Telescope.
