The scaling of local quantum entropies is of utmost interest for characterizing quantum fields, many-body systems, and gravity. Despite their importance, theoretically and experimentally accessing quantum entropies is challenging as they are nonlinear functionals of the underlying quantum state. Here, we show that suitably chosen classical entropies capture many features of their quantum analogs for an experimentally relevant setting. We describe the postquench dynamics of a multiwell spin-1 Bose-Einstein condensate from an initial product state via measurement distributions of spin observables and estimate the corresponding entropies using the asymptotically unbiased $k$-nearest-neighbor method. We observe the dynamical buildup of quantum correlations signaled by an area law, as well as local thermalization revealed by a transition to a volume law, both in regimes characterized by non-Gaussian distributions. We emphasize that all relevant features can be observed at small sample numbers without reconstructing the underlying state or measurement distributions, rendering our method directly applicable to a large variety of models and experimental platforms.

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• Bose gases
• Eigenstate thermalization
• Entanglement entropy
• Entanglement in quantum gases
• Entanglement production
• Entropy
• Entropy production
• Quantum entanglement
• Quantum quench
• Bose-Einstein condensates
• Quantum many-body systems
• Spinor Bose-Einstein condensates
• Ultracold gases