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Modern quantum simulators now routinely realize highly entangled many‑body dynamics, raising two related challenges: how to efficiently probe the resulting quantum states in experiment, and how to quantify how hard they are to simulate classically.

In the first part of the talk, I will discuss how randomized measurements and classical shadows provide a flexible toolbox to access nontrivial properties of quantum many‑body states, focusing on an experimentally friendly scheme to estimate von Neumann entropies via stabilized analytic continuation of shadow data. I will also describe optimized, gauge‑invariant shadow protocols for lattice gauge theories that exploit local constraints to efficiently access gauge‑invariant observables.

In the second part, I will discuss ongoing work on the classical simulation complexity of quantum dynamics via Heisenberg‑picture Pauli‑path methods. I will introduce a complexity metric - reactivity - which quantifies how strongly a quantum experiment as a whole responds to local perturbations: low reactivity implies efficiency for Pauli‑path simulations, whereas high reactivity flags regimes beyond these attacks. I will outline Pauli‑path spectroscopy, a simple protocol for measuring reactivity in the lab and guiding the search for complex quantum experiments.