  Radio pulsars provide some of the most important constraints for our
understanding of matter at supranuclear densities. So far, these constraints
are mostly given by precision mass measurements of neutron stars (NS). By
combining single measurements of the two most massive pulsars, J0348$+$0432 and
J0740$+$6620, the resulting lower limit of 1.98 $M_\odot$ (99% confidence) of
the maximum NS mass, excludes a large number of equations of state (EOSs).
Further EOS constraints, complementary to other methods, are likely to come
from the measurement of the moment of inertia (MOI) of binary pulsars in
relativistic orbits. The Double Pulsar, PSR J0737$-$3039A/B, is the most
promising system for the first measurement of the MOI via pulsar timing.
Reviewing this method, based in particular on the first MeerKAT observations of
the Double Pulsar, we provide well-founded projections into the future by
simulating timing observations with MeerKAT and the SKA. For the first time, we
account for the spin-down mass loss in the analysis. Our results suggest that
an MOI measurement with 11% accuracy (68% confidence) is possible by 2030. If
by 2030 the EOS is sufficiently well known, however, we find that the Double
Pulsar will allow for a 7% test of Lense-Thirring precession, or alternatively
provide a $\sim3\sigma$-measurement of the next-to-leading order gravitational
wave damping in GR. Finally, we demonstrate that potential new discoveries of
double NS systems with orbital periods shorter than that of the Double Pulsar
promise significant improvements in these measurements and the constraints on
NS matter.
