  We study the level-spacing statistics in the entanglement spectrum of output
states of random universal quantum circuits where qubits are subject to a
finite probability of projection to the computational basis at each time step.
We encounter two phase transitions with increasing projection rate: The first
is the volume-to-area law transition observed in quantum circuits with
projective measurements; The second separates the pure Poisson level statistics
phase at large projective measurement rates from a regime of residual level
repulsion in the entanglement spectrum within the area-law phase, characterized
by non-universal level spacing statistics that interpolates between the
Wigner-Dyson and Poisson distributions. By applying a tensor network
contraction algorithm introduced in Ref. [1] to the circuit spacetime, we
identify this second projective-measurement-driven transition as a percolation
transition of entangled bonds. The same behavior is observed in both circuits
of random two-qubit unitaries and circuits of universal gate sets, including
the set implemented by Google in its Sycamore circuits.
