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We present the result of the quadratic-in-spin interaction Hamiltonian forbinary systems of rotating compact objects with generic spins, up to NNNLOcorrections within the post-Newtonian expansion. The calculation is performedby employing the effective field theory diagrammatic approach, and it involvesFeynman integrals up to three loops, evaluated within the dimensionalregularization scheme. The gauge-invariant binding energy and the scatteringangle, in special kinematic regimes and spin configurations, are explicitlyderived. The results extend our earlier study on the spin-orbit interactioneffects.

Gravitational Quadratic-in-Spin Hamiltonian at NNNLO in the post-Newtonian framework