A study of the horn in the particle ratio $K^+/pi^+$ for central heavy-ion collisions as a function of the collision energy $sqrt{s}$ is presented. We analyse two different interpretations: the onset of deconfinement and the transition from a baryon-
to a meson-dominated hadron gas. We use a realistic equation of state (EOS), which includes both hadron and quark degrees-of-freedom. The Taub-adiabate procedure is followed to determine the system at the early stage. Our results do not support an explanation of the horn as due to the onset of deconfinement. Using only hadronic EOS we reproduced the energy dependence of the $K^+/pi^+$ and $Lambda/pi^-$ ratios employing an experimental parametrisation of the freeze-out curve. We observe a transition between a baryon- and a meson-dominated regime; however, the reproduction of the $K^+/pi^+$ and $Lambda/pi^-$ ratios as a function of $sqrt{s}$ is not completely satisfying. We finally propose a new idea for the interpretation of the data, the roll-over scheme, in which the scalar meson field $sigma$ has not reached the thermal equilibrium at freeze-out. The rool-over scheme for the equilibration of the $sigma$-field is based on the inflation mechanism. The non-equilibrium evolution of the scalar field influences the particle production, e.g. $K^+/pi^+$, however, the fixing of the free parameters in this model is still an open issue.
The implications of the formation of strange quark matter in neutron stars and in core-collapse supernovae is discussed with special emphasis on the possibility of having a strong first order QCD phase transition at high baryon densities. If strange
quark matter is formed in core-collapse supernovae shortly after the bounce, it causes the launch of a second outgoing shock which is energetic enough to lead to a explosion. A signal for the formation of strange quark matter can be read off from the neutrino spectrum, as a second peak in antineutrinos is released when the second shock runs over the neutrinosphere.
The effect of the QCD phase transition is studied for the mass-radius relation of compact stars and for hot and dense matter at a given proton fraction used as input in core-collapse supernova simulations. The phase transitions to the 2SC and CFL col
or superconducting phases lead to stable hybrid star configurations with a pure quark matter core. In supernova explosions quark matter could be easily produced due to $beta$-equilibrium, small proton fractions and nonvanishing temperatures. A low critical density for the phase transition to quark matter is compatible with present pulsar mass measurements.
