Termwise versus globally stoquastic local Hamiltonians: questions of complexity and sign-curing

We elucidate the distinction between global and termwise stoquasticity for local Hamiltonians and prove several complexity results. We show that the stoquastic local Hamiltonian problem is $\textbf{StoqMA}$-complete even for globally stoquastic Hamiltonians. We study the complexity of deciding whether a local Hamiltonian is globally stoquastic or not. In particular, we prove $\textbf{coNP}$-hardness of deciding global stoquasticity in a fixed basis and $\Sigma_2^p$-hardness of deciding global stoquasticity under single-qubit transformations. As a last result, we expand the class of sign-curing transformations by showing how Clifford transformations can sign-cure a class of disordered 1D $XYZ$ Hamiltonians.