Recent rapid progress in multimessenger observations of neutron stars (NSs) offers great potential to constrain the properties of strongly interacting matter under the most extreme conditions. In order to fully exploit the current observational inputs and to study the impact of future observations, we analyze a large ensemble of randomly generated model-independent equations of state (EoSs) and the corresponding rotating stellar structures without the use of quasi-universal relations. We discuss the compatibility and impact of various hypotheses and measurements on the EoS, including those involving the merger product in GW170817, the binary merger components in GW190814, and radius measurements of PSR J0740+6620. We obtain an upper limit for the dimensionless spin of a rigidly rotating NS, |chi| < 0.81, an upper limit for the compactness of a NS, GM/(Rc^2) < 0.33, and find that the conservative hypothesis that the remnant in GW170817 ultimately collapsed to a black hole strongly constrains the EoS and the maximal mass of NSs, implying M_TOV < 2.53M_sol (or M_TOV < 2.19M_sol if we assume that a hypermassive NS was created). Furthermore, we find that the recent NICER results for the radius of the massive PSR J0740+6620 offer strong constraints for the EoS, and that the indicated radius values for a two-solar mass NS greater than about 11 km are completely compatible with the presence of quark matter in massive NSs.
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