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We address how one can empirically infer properties of quantum statesgenerated by dynamics involving measurements. Our focus is on many-bodysettings where the number of measurements is extensive, making brute-forceapproaches based on postselection intractable due to their exponential samplecomplexity. We introduce a general-purpose inference scheme for learningproperties of the post-measurement ensemble of states using a scalable numberof experimental repetitions. We first identify a general class of `estimableproperties' that can be directly extracted from experimental data. Then, basedon empirical observations of some such quantities, we show how one canindirectly infer information about a given non-estimable quantity of interest,such as the average entanglement entropy, or frame potential. We formulate ourapproach in terms of an optimization task, where one asks what are the minimumand maximum values that the desired quantity could possibly take, whileensuring consistency with observations. The true value of this quantity mustthen lie within a feasible range between these extrema, resulting in two-sidedbounds. Narrow feasible ranges can be obtained by using a classical simulationof the device to determine which estimable properties one should measure. Evenin cases where this simulation is inaccurate, unambiguous information about thetrue 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 theemergence of quantum state designs in experiments. We identify some fundamentalobstructions that in some cases prevent sharp knowledge of a given quantityfrom being inferred, and discuss what can be learned in cases where classicalsimulation is too computationally demanding to be feasible.

Postselection-free learning of measurement-induced quantum dynamics