  Recent rapid progress in neutron-star (NS) observations 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 of NS masses, radii, and
tidal deformabilities, we analyze a large ensemble of randomly generated viable
NS-matter equations of state (EoSs) and the corresponding rotating stellar
structures. We discuss the compatibility and impact of various hypotheses and
measurements on the EoS, including those involving the merger product of the
gravitational-wave (GW) event GW170817, the binary merger components in
GW190814, and radius measurements of the pulsar 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 $M_\text{TOV} < 2.53 M_\odot$ (or $M_\text{TOV} < 2.19 M_\odot$
if we assume that a hypermassive NS was created). Additionally, we derive a
novel lower limit for the tidal deformability as a function of NS mass and
provide fitting formulae that can be used to set priors for parameter
estimation and to discern whether neutron stars or other compact objects are
involved in future low-mass GW events. Finally, we find that the recent NICER
results for the radius of the massive NS PSR J0740+6620 place strong
constraints for the behavior of the EoS, and that the indicated radius values
$R(2M_\odot) \gtrsim 11$ km are compatible with moderate speeds of sound in NS
matter and thus with the existence of quark matter cores in massive NSs.
