  We address how one can empirically infer properties of quantum states
generated by dynamics involving measurements. Our focus is on many-body
settings where the number of measurements is extensive, making brute-force
approaches based on postselection intractable due to their exponential sample
complexity. We introduce a general-purpose scheme that can be used to infer any
property of the post-measurement ensemble of states (e.g. the average
entanglement entropy, or frame potential) using a scalable number of
experimental repetitions. We first identify a general class of estimable
properties that can be directly extracted from experimental data. Then, based
on empirical observations of such quantities, we show how one can indirectly
infer information about any particular given non-estimable quantity of interest
through classical post-processing. Our approach is based on an optimization
task, where one asks what are the minimum and maximum values that the desired
quantity could possibly take, while ensuring consistency with observations. The
true value of this quantity must then lie within a feasible range between these
extrema, resulting in two-sided bounds. Narrow feasible ranges can be obtained
by using a classical simulation of the device to determine which estimable
properties one should measure. Even in cases where this simulation is
inaccurate, unambiguous information about the true value of a given quantity
realised on the quantum device can be learned. As an immediate application, we
show that our method can be used to verify the emergence of quantum state
designs in experiments. We identify some fundamental obstructions that in some
cases prevent sharp knowledge of a given quantity from being inferred, and
discuss what can be learned in cases where classical simulation is too
computationally demanding to be feasible.
