  The next generation of ground-based gravitational-wave detectors, Einstein
Telescope (ET) and Cosmic Explorer (CE), present a unique opportunity to put
constraints on dense matter, among many other groundbreaking scientific goals.
In a recent study the science case of ET was further strengthened, studying in
particular the performances of different detector designs. In this paper we
present a more detailed study of the nuclear physics section of that work. In
particular, focusing on two different detector configurations (the single-site
triangular-shaped design and a design consisting of two widely separated
"L-shaped" interferometers), we study the detection prospects of binary neutron
star (BNS) mergers, and how they can reshape our understanding of the
underlying equation of state (EoS) of dense matter. We employ several
state-of-the-art EoS models and state-of-the-art synthetic BNS merger catalogs,
and we make use of the Fisher information formalism (FIM) to quantify
statistical errors on the astrophysical parameters describing individual BNS
events. To check the reliability of the FIM method, we further perform a full
parameter estimation for a few simulated events. Based on the uncertainties on
the tidal deformabilities associated to these events, we outline a mechanism to
extract the underlying injected EoS using a recently developed meta-modelling
approach within a Bayesian framework. Our results suggest that with $\gtrsim
500$ events with signal-to-noise ratio greater than $12$, we will be able to
pin down very precisely the underlying EoS governing the neutron star matter.
