  We establish the emergence of a conformal field theory (CFT) in a
(1+1)-dimensional hybrid quantum circuit right at the measurement-driven
entanglement transition by revealing space-time conformal covariance of
entanglement entropies and mutual information for various subregions at
different circuit depths. While the evolution takes place in real time, the
spacetime manifold of the circuit appears to host a Euclidean field theory with
imaginary time. Throughout the paper we investigate Clifford circuits with
several different boundary conditions by injecting physical qubits at the
spatial and/or temporal boundaries, all giving consistent characterizations of
the underlying "Clifford CFT." We emphasize (super)universal results that are
consequences solely of the conformal invariance and do not depend crucially on
the precise nature of the CFT. Among these are the infinite entangling speed as
a consequence of measurement-induced quantum nonlocality and the critical
purification dynamics of a mixed initial state.
