Entanglement is one of the key features of the quantum world, which, over the past decades, has been promoted to a unifying concept in exploring diverse phenomena ranging from black hole physics to condensed matter systems. Particularly, understanding entanglement and its dynamics has laid modern foundations for manybody physics. Unfortunately, directly probing entanglement in experiments requires very delicate, system-dependent tools whose complexity grows exponentially with system size. We argue that in systems with conservation laws, fluctuations of conserved extensive quantum observables provide an alternative route to probe entanglement and its scaling laws. This leads to an exponential reduction in the resources needed to measure entanglement in comparison to the direct method known as quantum state tomography. Our heuristic statistical arguments suggesting fluctuations as a probe for entanglement are supported with numerical simulations of a variety of systems including ground states of non-interacting fermionic models and interacting spin models. More interestingly, we show that entanglement and fluctuations behave hand in hand in "monitored quantum circuits" where the interplay of unitary dynamics governed by local interactions and non-unitary quantum measurements leads to a very rich phase diagram for entanglement.
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