  Boundary-driven spin chains are paradigmatic non-equilibrium systems in both
classical and quantum settings. In general it may not be possible to
distinguish classical from quantum transport through monitoring the mean
current, as both ballistic as well as diffusive regimes occur in either
setting. Here we show that genuine quantum features become manifest in large
fluctuations which allow a discrimination between classical and quantum
transport: in the classical case, realizations that are characterized by
atypically large boundary activity are associated with larger than typical
currents, i.e. an enhanced number of events at the boundaries goes together
with a large current. Conversely, in the quantum case the Zeno effect leads to
the suppression of current in trajectories with large activity at the boundary.
We analyze how these different dynamical regimes are reflected in the structure
of rare fluctuations. We show furthermore that realizations supporting a large
current are generated via weak long-range correlations within the spin chain,
typically associated with hyperuniformity.
