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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. 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.