  The interplay between unitary dynamics and local quantum measurements results
in unconventional non-unitary dynamical phases and transitions. In this paper
we investigate the dynamics of $(d+1)$-dimensional hybrid stabilizer circuits,
for $d=1,2,3$. We characterize the measurement-induced phases and their
transitions using large-scale numerical simulations focusing on entanglement
measures, purification dynamics, and wave-function structure. Our findings
demonstrate the measurement-induced transition in $(d+1)$ spatiotemporal
dimensions is conformal and close to the percolation transition in $(d+1)$
spatial dimensions.
