  We explore QCD$_4$ quark matter, the $\mu$-T (chemical potential-temperature)
phase diagram, possible 't Hooft anomalies, and topological terms, via
non-perturbative tools of cobordism theory and higher anomaly matching. We
focus on quarks in 3-color and 3-flavor on bi-fundamentals of SU(3), then
analyze the continuous and discrete global symmetries and pay careful attention
to finite group sectors. We input constraints from $T=CP$ or $CT$ time-reversal
symmetries, implementing QCD on unorientable spacetimes and distinct topology.
Examined phases include the high T QGP (quark-gluon plasma/liquid), the low T
ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL
(3-color-flavor locking superconductivity) at high density. We introduce a
possibly useful but only approximate higher anomaly, involving discrete 0-form
axial and 1-form mixed chiral-flavor-locked center symmetries, matched by the
above four QCD phases. We also enlist as much as possible, but without
identifying all of, 't Hooft anomalies and topological terms relevant to
Symmetry Protected/Enriched Topological states (SPTs/SETs) of gauged SU(2) or
SU(3) QCD$_d$-like matter theories in general in any spacetime dimensions
$d=2,3,4,5$ via cobordism.
