  We study measurement-induced entanglement and control phase transitions in a
quantum analog of the Bernoulli map subjected to a classically-inspired control
protocol. When entangling gates are restricted to the Clifford group, separate
entanglement ($p_\mathrm{ent}$) and control ($p_\mathrm{ctrl}$) transitions
emerge, revealing two distinct universality classes. The control transition has
critical exponents $\nu$ and $z$ consistent with the classical map (a random
walk) while the entanglement transition is revealed to have similar exponents
as the measurement-induced phase transition in Clifford hybrid dynamics. This
is distinct from the case of generic entangling gates in the same model, where
$p_\mathrm{ent} = p_\mathrm{ctrl}$ and universality is controlled by the random
walk.
