  We introduce the concepts of a symmetry-protected sign problem and
symmetry-protected magic to study the complexity of symmetry-protected
topological (SPT) phases of matter. In particular, we say a state has a
symmetry-protected sign problem or symmetry-protected magic, if finite-depth
quantum circuits composed of symmetric gates are unable to transform the state
into a non-negative real wave function or stabilizer state, respectively. We
prove that states belonging to certain SPT phases have these properties, as a
result of their anomalous symmetry action at a boundary. For example, we find
that one-dimensional $\mathbb{Z}_2 \times \mathbb{Z}_2$ SPT states (e.g.
cluster state) have a symmetry-protected sign problem, and two-dimensional
$\mathbb{Z}_2$ SPT states (e.g. Levin-Gu state) have symmetry-protected magic.
Furthermore, we comment on the relation between a symmetry-protected sign
problem and the computational wire property of one-dimensional SPT states. In
an appendix, we also introduce explicit decorated domain wall models of SPT
phases, which may be of independent interest.
