Dense nuclear matter is expected to be a superfluid. Quark matter, which is expected to appear at higher densities, is also a superfluid. But quark matter turns out to be sharply distinct from a standard superfluid, because it supports Z3-valued particle-vortex braiding phases, and its effective action includes a coupling to a topological quantum field theory. Physically, our results imply that certain mesonic and baryonic excitations have orbital angular momentum quantized in units of ħ/3 in the presence of a superfluid vortex. If Z3 braiding phases and angular momentum fractionalization are absent in lower density hadronic matter, as is widely expected, then the quark matter and hadronic matter regimes of dense QCD must be separated by at least one phase transition. Since the low-density regime is a `confined' phase, while the high-density regime is a `Higgs' phase due to color superconductivity, our results also have interesting implications for Higgs-confinement complementarity.