  From the S-matrix of spinning particles, we extract the 2 PM conservative
potential for binary spinning black holes up to quartic order in spin
operators. An important ingredient is the exponentiated gravitational Compton
amplitude in the classical spin-limit for all graviton helicity sectors. The
validity of the resulting Hamiltonian is verified by matching to known lower
spin order results, as well as direct computation of the 2PM impulse and spin
kicks from the eikonal phase and that from the test black hole scattering based
on Mathisson-Papapetrou-Dixon equations.
