We have developed recently protocols based on statistical correlations between random measurements in context of engineered quantum many-body systems [1-5]. Here we first review application of these protocols, which range from measurement of Renyi entanglement entropies [1,2], Out-Of-Time-Ordered Correlators [3] and Topological Invariants of Many-Body Systems [4]. In this talk we focus in particular on Cross-Platform Verification of quantum computers and simulators [5]. We show how to measure the overlap between (reduced) density matrices, and the (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. As a proof-of-principle, we present the measurement of experiment-theory fidelities for entangled 10-qubit quantum states in a trapped ion quantum simulator.  [1] A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller Phys. Rev. Lett. 120, 050406 (2018) [2] T. Brydges,, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B.P. Lanyon, P. Zoller, R. Blatt, C. F. Roos, Science 364, 260 (2019) [3] B. Vermersch, A. Elben, L. M. Sieberer, N. Y. Yao, and P. Zoller Phys. Rev. X 9, 021061 (2019) [4] A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, B. Vermersch, arXiv:1906.05011 [5] A. Elben, B. Vermersch, R. van Bijnen, C. Kokail, T. Brydges, C. Maier, M Joshi, R. Blatt, C. F. Roos, P.Zoller, to be submitted

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Quantum many-body systems are very hard to simulate, as computational resources (time and memory) typically grow exponentially with system size. However, quantum computers or analog quantum simulators may perform that task in a much more efficient way. In this talk, I will first review some of the methods that have been proposed for this task and then explain the advantages and disadvantages of analog quantum simulators with respect to quantum computers. I will also describe possible applications in condensed matter physics, high-energy physics, and chemistry.