  One of the primary science goals of the next generation of hard X-ray timing
instruments is to determine the equation of state of the matter at supranuclear
densities inside neutron stars, by measuring the radius of neutron stars with
different masses to accuracies of a few percent. Three main techniques can be
used to achieve this goal. The first involves waveform modelling. The flux we
observe from a hotspot on the neutron star surface offset from the rotational
pole will be modulated by the star's rotation, giving rise to a pulsation.
Information about mass and radius is encoded into the pulse profile via
relativistic effects, and tight constraints on mass and radius can be obtained.
The second technique involves characterising the spin distribution of accreting
neutron stars. The most rapidly rotating stars provide a very clean constraint,
since the mass-shedding limit is a function of mass and radius. However the
overall spin distribution also provides a guide to the torque mechanisms in
operation and the moment of inertia, both of which can depend sensitively on
dense matter physics. The third technique is to search for quasi-periodic
oscillations in X-ray flux associated with global seismic vibrations of
magnetars (the most highly magnetized neutron stars), triggered by magnetic
explosions. The vibrational frequencies depend on stellar parameters including
the dense matter equation of state. We illustrate how these complementary X-ray
timing techniques can be used to constrain the dense matter equation of state,
and discuss the results that might be expected from a 10m$^2$ instrument. We
also discuss how the results from such a facility would compare to other
astronomical investigations of neutron star properties. [Modified for arXiv]
