Recently, protocols based on statistical correlations of randomized measurements were developed for probing and verifying engineered quantum many-body systems. After a general introduction in the context of Renyi entropies, I focus in this talk on the cross-platform verification of quantum computers and simulators by means of fidelity measurements. I show how to measure the overlap between (reduced) density matrices, and thus a (mixed-state) fidelity of two quantum states prepared on separate experimental platforms. The protocol requires only local measurements in randomized product bases and classical communication between the devices. As a proof-of-principle, I present the measurement of experiment-theory fidelities for entangled 10-qubit quantum states in a trapped ion quantum simulator. To conclude, I will present further applications of randomized measurements for probing quantities beyond standard observables, such as out-of-time-ordered correlation functions.