  We examine the problem of determining whether a multi-qubit two-local
Hamiltonian can be made stoquastic by single-qubit unitary transformations. We
prove that when such a Hamiltonian contains one-local terms, then this task can
be NP-hard. This is shown by constructing a class of Hamiltonians for which
performing this task is equivalent to deciding $3$-SAT. In contrast, we show
that when such a Hamiltonian contains no one-local terms then this task is
easy, namely we present an algorithm which decides, in a number of arithmetic
operations over $\mathbb{R}$ which is polynomial in the number of qubits,
whether the sign problem of the Hamiltonian can be cured by single-qubit
rotations.
