  In this paper we derive for the first time the complete gravitational
quartic-in-spin interaction of generic compact binaries at the next-to-leading
order in the post-Newtonian (PN) expansion. The derivation builds on the
effective field theory for gravitating spinning objects, and its recent
extensions, in which new type of worldline couplings should be considered, as
well as on the extension of the effective action of a spinning particle to
quadratic order in the curvature. The latter extension entails a new Wilson
coefficient that appears in this sector. This work pushes the precision
frontier with spins at the fifth PN (5PN) order for maximally-spinning compact
objects, and at the same time informs us of the gravitational Compton
scattering with higher spins.
