We explore the implications of the QCD phase transition during the postbounce evolution of core-collapse supernovae. Using the MIT bag model for the description of quark matter, we model phase transitions that occur during the early postbounce evolution. This stage of the evolution can be simulated with general relativistic three-flavor Boltzmann neutrino transport. The phase transition produces a second shock wave that triggers a delayed supernova explosion. If such a phase transition happens in a future galactic supernova, its existence and properties should become observable as a second peak in the neutrino signal that is accompanied by significant changes in the energy of the emitted neutrinos. This second neutrino burst is dominated by the emission of antineutrinos because the electron degeneracy is reduced when the second shock passes through the previously neutronized matter.

We discuss the formation of $$\Delta${}$ isobars in neutron star matter. We show that their threshold density strictly correlates with the density derivative of the symmetry energy of nuclear matter: the $L$ parameter. By restricting $L$ to the range of values indicated by recent experimental and theoretical analysis, i.e., $40\phantom{\rule{4.pt}{0ex}}\text{MeV}$\lesssim${}L$\lesssim${}62\phantom{\rule{4.pt}{0ex}}\text{MeV}$, we find that $$\Delta${}$ isobars appear at a density of the order of 2 to 3 times the nuclear matter saturation density, i.e., the same range as for the appearance of hyperons. The range of values of the couplings of the $$\Delta${}\mathrm{s}$ with the mesons is restricted by the analysis of the data obtained from photoabsorption, electron and pion scattering on nuclei. If the potential of the $$\Delta${}$ in nuclear matter is close to the one indicated by the experimental data then the equation of state becomes soft enough that a ``$\Delta${} puzzle'' exists, similar to the ``hyperon puzzle'' widely discussed in the literature. Possible solutions to this puzzle are also discussed.