Recent astrophysical measurements of compact star (CS) masses, mass radius relations and their cooling evolution give valuable insight into the properties of cold and dense nuclear matter (NM) beyond saturation. These data provide a benchmark for our understanding of the high density equation of state (EoS). Additionally the EoS has to be in accordance with flow data analyses for isospin symmetric matter (SM). Applying these constraints results in a highly selective testing scheme on the nuclear EoS. This scheme allows in a next step to model a hybrid, nuclear-quark matter EoS by chosing phenomenologically reliable parameter sets for a NJL-type quark matter model EoS. The resulting hybrid star configurations are barely distinguishable from pure neutron stars. A possible consequence for the QCD phase diagram is an almost cross over transition from quark matter to nuclear matter. The last part of the talk gives an outlook to the development of an EoS which shall be applicable for, e.g., supernovae and neutron star merger simulations, taking into account the presence of light and heavy elements.
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