  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.
