  We compare two classes of hybrid equations of state with a hadron-to-quark
matter phase transition in their application to core collapse supernova
simulations. The first one uses the quark bag model and describes the
transition to three-flavor quark matter at low critical densities. The second
one employs a Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model with
parameters describing a phase transition to two-flavor quark matter at higher
critical densities. These models possess a distinctly different temperature
dependence of their transition densities which turns out to be crucial for the
possible appearance of quark matter in supernova cores. During the early post
bounce accretion phase quark matter is found only if the phase transition takes
place at sufficiently low densities as in the study based on the bag model. The
increase critical density with increasing temperature, as obtained for our PNJL
parametrization, prevents the formation of quark matter. The further evolution
of the core collapse supernova as obtained applying the quark bag model leads
to a structural reconfiguration of the central proto-neutron star where, in
addition to a massive pure quark matter core, a strong hydrodynamic shock wave
forms and a second neutrino burst is released during the shock propagation
across the neutrinospheres. We discuss the severe constraints in the freedom of
choice of quark matter models and their parametrization due to the recently
observed 2 solar mass pulsar and their implications for further studies of core
collapse supernovae in the QCD phase diagram.
