  We show that $\mathbb{Z}_3$-valued particle-vortex braiding phases are
present in high density quark matter. Certain mesonic and baryonic excitations,
in the presence of a superfluid vortex, have orbital angular momentum quantized
in units of $\hbar/3$. Such non-local topological features can distinguish
phases whose realizations of global symmetries, as probed by local order
parameters, are identical. If $\mathbb{Z}_3$ 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.
