Among the most robust phenomena characterizing systems composed of large numbers of constituents is the emergence of normal fluctuations around the mean. However, in semi-classical limits of strongly interacting many-body quantum systems, anomalous fluctuations have been observed, indicating nontrivial extensions of central-limit behavior to correlated systems. Understanding such universal behavior, probed in recent experiments using state-of-the-art quantum simulators, is a crucial step towards identifying, characterizing, and controlling emergent phenomena that will underpin novel quantum computing technologies in line with Cluster 4 of the Horizon Europe Work Programme. The project FAQs – Fluctuation Anomaly of Quantum Systems will advance the state of the art concerning fluctuations of quantum systems using analytical and numerical techniques. A multifaceted understanding of quantum fluctuations will be obtained by combining ideas and techniques from hydrodynamics, integrable probability, and exact solutions of minimal quantum models. The project will first elucidate the hydrodynamic origin of anomalous fluctuations in strongly interacting quantum systems and describe the resultant universality class. To achieve a deeper understanding of quantum dynamics, the microscopic origin of convective diffusion will be worked out exactly by applying the tools of integrable combinatorics to quantum spin chains. Finally, the project will address the important open problem of partial Kardar-Parisi-Zhang physics in the dynamics of the Heisenberg spin chain.