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We show that 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 1/3. Such non-local topological features can distinguish phases whose realizations of global symmetries, as probed by local order parameters, are identical. If 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.