  We study the problem of observing quantum collective phenomena emerging from
large numbers of measurements. These phenomena are difficult to observe in
conventional experiments because, in order to distinguish the effects of
measurement from dephasing, it is necessary to post-select on sets of
measurement outcomes whose Born probabilities are exponentially small in the
number of measurements performed. An unconventional approach, which avoids this
exponential `post-selection problem', is to construct cross-correlations
between experimental data and the results of simulations on classical
computers. However, these cross-correlations generally have no definite
relation to physical quantities. We first show how to incorporate shadow
tomography into this framework, thereby allowing for the construction of
quantum information-theoretic cross-correlations. We then identify
cross-correlations which both upper and lower bound the measurement-averaged
von Neumann entanglement entropy, and additional cross-correlations which lower
bound the measurement-averaged purity and entanglement negativity. These bounds
show that experiments can be performed to constrain post-measurement
entanglement without the need for post-selection. To illustrate our technique
we consider how it could be used to observe the measurement-induced
entanglement transition in Haar-random quantum circuits. We use exact numerical
calculations as proxies for quantum simulations and, to highlight the
fundamental limitations of classical memory, we construct cross-correlations
with tensor-network calculations at finite bond dimension. Our results reveal a
signature of measurement-induced criticality that can be observed using a
quantum simulator in polynomial time and with polynomial classical memory.
