  We study the complexity of QMA proof systems with inverse exponentially small
promise gap. We show that this class can be exactly characterized by PSPACE,
the class of problems solvable with a polynomial amount of memory. As
applications we show that a "precise" version of the Local Hamiltonian problem
is PSPACE-complete, and give a provable setting in which the ability to prepare
PEPS states is not as powerful as the ability to prepare the ground state of
general Local Hamiltonians.
