Quantum Merlin-Arthur and Proofs Without Relative Phase

We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [21]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu [24]) can be much more powerful. We show that QMA(+) with some constant gap is equal to NEXP, yet QMA(+) with some other constant gap is equal to QMA. One interpretation is that Merlin's ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2)subset of NEXP.