  It is currently unknown how matter behaves at the extreme densities found
within the cores of neutron stars. Measurements of the neutron star equation of
state probe nuclear physics that is otherwise inaccessible in a laboratory
setting. Gravitational waves from binary neutron star mergers encode details
about this physics, allowing the equation of state to be inferred. Planned
third-generation gravitational-wave observatories, having vastly improved
sensitivity, are expected to provide tight constraints on the neutron star
equation of state. We combine simulated observations of binary neutron star
mergers by the third-generation observatories Cosmic Explorer and Einstein
Telescope to determine future constraints on the equation of state across a
plausible neutron star mass range. In one year of operation, a network
consisting of one Cosmic Explorer and the Einstein Telescope is expected to
detect $\gtrsim 3\times 10^5$ binary neutron star mergers. By considering only
the 75 loudest events, we show that such a network will be able to constrain
the neutron star radius to at least $\lesssim 200$ m (90% credibility) in the
mass range $1-1.97$ $M_{\odot}$ -- about ten times better than current
constraints from LIGO-Virgo-KAGRA and NICER. The constraint is $\lesssim 75$ m
(90% credibility) near $1.4-1.6$ $M_{\odot}$ where we assume the the binary
neutron star mass distribution is peaked. This constraint is driven primarily
from the loudest $\sim 20$ events.
