With the detection of GW170817 we have observed the first multi-messenger gravitational wave signal from two merging neutron stars. This signal carried a multitude of information about the underlying equation of state (EOS) of nuclear matter, which so far is not known for densities above nuclear saturation. In particular it is not known if exotic states or even a phase transition to quark matter can occur at densities so extreme that they cannot be probed by any ground-based experiment. I will show how the information carried in the gravitational wave signal of GW170817 can be used to constrain the EOS at densities above saturation and what we can learn about the possible existence of phase transitions. I will also comment on how we can improve on those limits with upcoming observations of the NICER mission.
In the second part of the talk, I will focus on yet to be observed electromagnetic precursor emission from neutron star mergers. These precursors have the potential to provide complementary insights into the properties of neutron stars, such as their spins, which cannot yet be reliably extracted from the gravitational wave signal. I will present preliminary results on the interaction of force-free magnetospheres in compact binaries and demonstrate how powerful electromagnetic flares can be launched for a wide variety of orbital parameters.