  We discuss an idea for how accreting millisecond pulsars could contribute to
the understanding of the QCD phase transition in the high-density nuclear
matter equation of state (EoS). It is based on two ingredients, the first one
being a ``phase diagram'' of rapidly rotating compact star configurations in
the plane of spin frequency and mass, determined with state-of-the-art hybrid
equations of state, allowing for a transition to color superconducting quark
matter. The second is the study of spin-up and accretion evolution in this
phase diagram. We show that the quark matter phase transition leads to a
characteristic line in the Omega-M plane, the phase border between neutron
stars and hybrid stars with a quark matter core. Along this line a change in
the pulsar's moment of inertia entails a waiting point phenomenon in the
accreting millisecond X-ray pulsar (AMXP) evolution: most of these objects
should therefore be found along the phase border in the Omega-M plane, which
may be viewed as the AMXP analog of the main sequence in the
Hertzsprung-Russell diagram for normal stars. In order to prove the existence
of a high-density phase transition in the cores of compact stars we need
population statistics for AMXP's with sufficiently accurate determination of
their masses and spin frequencies.
