  The Local Hamiltonian problem (finding the ground state energy of a quantum
system) is known to be QMA-complete. The Local Consistency problem (deciding
whether descriptions of small pieces of a quantum system are consistent) is
also known to be QMA-complete. Here we consider special cases of Local
Hamiltonian, for ``stoquastic'' and 1-dimensional systems, that seem to be
strictly easier than QMA. We show that there exist analogous special cases of
Local Consistency, that have equivalent complexity (up to poly-time oracle
reductions). Our main technical tool is a new reduction from Local Consistency
to Local Hamiltonian, using SDP duality.
