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

The Local Consistency Problem for Stoquastic and 1-D Quantum Systems