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We elucidate the distinction between global and termwise stoquasticity forlocal Hamiltonians and prove several complexity results. We show that thestoquastic local Hamiltonian problem is $\textbf{StoqMA}$-complete even forglobally stoquastic Hamiltonians. We study the complexity of deciding whether alocal 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-qubittransformations. As a last result, we expand the class of sign-curingtransformations by showing how Clifford transformations can sign-cure a classof disordered 1D $XYZ$ Hamiltonians.

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