Transport measurements on the cuprates suggest the presence of a quantum critical point (QCP) hiding underneath the superconducting dome near optimal hole doping. We provide numerical evidence in support of this scenario via a dynamical cluster quantum Monte Carlo study of the extended two-dimensional Hubbard model. Single-particle quantities, such as the spectral function, the quasi-particle weight and the entropy, display a crossover between two distinct ground states: a Fermi liquid at low filling and a non-Fermi liquid with a pseudo-gap at high filling. Both states are found to cross over to a marginal Fermi-liquid state at higher temperatures. For finite next-nearest-neighbour hopping t′, we find a classical critical point at temperature Tc. This classical critical point is found to be associated with a phase-separation transition between a compressible Mott gas and an incompressible Mott liquid corresponding to the Fermi liquid and the pseudo-gap state, respectively. Since the critical temperature Tc extrapolates to zero as t′ vanishes, we conclude that a QCP connects the Fermi liquid to the pseudo-gap region, and that the marginal Fermi-liquid behaviour in its vicinity is the analogue of the supercritical region in the liquid–gas transition.