  In gauge theories with fundamental matter there is typically no sharp way to
distinguish confining and Higgs regimes, e.g. using generalized global
symmetries acting on loop order parameters. It is standard lore that these two
regimes are continuously connected, as has been explicitly demonstrated in
certain lattice and continuum models. We point out that Higgsing and
confinement sometimes lead to distinct symmetry protected topological (SPT)
phases -- necessarily separated by a phase transition -- for ordinary global
symmetries. We present explicit examples in 3+1 dimensions, obtained by adding
elementary Higgs fields and Yukawa couplings to QCD while preserving parity P
and time reversal T. In a suitable scheme, the confining phases of these
theories are trivial SPTs, while their Higgs phases are characterized by
non-trivial P- and T-invariant theta-angles $\theta_f, \theta_g = \pi$ for
flavor or gravity background gauge fields, i.e. they are topological insulators
or superconductors. Finally, we consider conventional three-flavor QCD (without
elementary Higgs fields) at finite $U(1)_B$ baryon-number chemical potential
$\mu_B$, which preserves P and T. At very large $\mu_B$, three-flavor QCD is
known to be a completely Higgsed color superconductor that also spontaneously
breaks $U(1)_B$. We argue that this high-density phase is in fact a gapless
SPT, with a gravitational theta-angle $\theta_g = \pi$ that safely co-exists
with the $U(1)_B$ Nambu-Goldstone boson. We explain why this SPT motivates
unexpected transitions in the QCD phase diagram, as well as anomalous surface
modes at the boundary of quark-matter cores inside neutron stars.
