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We explore QCD$_4$ quark matter, the $\mu$-T (chemical potential-temperature)phase diagram, possible 't Hooft anomalies, and topological terms, vianon-perturbative tools of cobordism theory and higher anomaly matching. Wefocus on quarks in 3-color and 3-flavor on bi-fundamentals of SU(3), thenanalyze the continuous and discrete global symmetries and pay careful attentionto finite group sectors. We input constraints from $T=CP$ or $CT$ time-reversalsymmetries, implementing QCD on unorientable spacetimes and distinct topology.Examined phases include the high T QGP (quark-gluon plasma/liquid), the low TChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL(3-color-flavor locking superconductivity) at high density. We introduce apossibly useful but only approximate higher anomaly, involving discrete 0-formaxial and 1-form mixed chiral-flavor-locked center symmetries, matched by theabove four QCD phases. We also enlist as much as possible, but withoutidentifying all of, 't Hooft anomalies and topological terms relevant toSymmetry Protected/Enriched Topological states (SPTs/SETs) of gauged SU(2) orSU(3) QCD$_d$-like matter theories in general in any spacetime dimensions$d=2,3,4,5$ via cobordism.

Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew