QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is “Consistency of Local Density Matrices”: given a collection of density matrices describing different subsets of an n-qubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, “Fermionic Local Hamiltonian” and “N-representability, ” are QMA-complete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. N-representability is particularly interesting, as it is a key component