Perhaps the most important problem in physics or quantum chemistry is to determine properties of the ground state of an interacting system of fermions. As a quantum mechanical problem, there may be no efficient classical witness to the ground state energy, or even to an approximation of that energy. A commonly considered witness is a so-called “Gaussian state”, or free fermion wavefunction. As a prominent example , the Sachdev-Ye-Kitaev (SYK) model has no Gaussian state which achieves a good approximation to the energy; this model is sometimes considered as one of the “most entangled” or “most strongly interacting” models possible. I will discuss applications of the sum-of-squares method to this model. Sum-of-squares is a semidefinite programming relaxation. I will show that this method can give classically efficient constant-factor lower bounds on the energy, and it inspires a quantum algorithm which gives constant-factor upper bounds. Joint work with R. O’Donnell.