  We prove a deterministic exponential time upper bound for Quantum
Merlin-Arthur games with k unentangled provers. This is the first non-trivial
upper bound of QMA(k) better than NEXP and can be considered an exponential
improvement, unless EXP=NEXP. The key ideas of our proof are to use
perturbation theory to reduce the QMA(2)-complete Separable Sparse Hamiltonian
problem to a variant of the Separable Local Hamiltonian problem with an
exponentially small promise gap, and then to decide this instance using
epsilon-net methods. Our results imply an exponential time algorithm for the
Pure State N-Representability problem in quantum chemistry, which is in QMA(2),
but is not known to be in QMA. We also discuss the implications of our results
on the Best Separable State problem.
