It is generally assumed within the standard cosmological model that initial
density perturbations are Gaussian at all scales. However, primordial quantum
diffusion unavoidably generates non-Gaussian, exponential tails in the
distribution of inflationary perturbations. These exponential tails have direct
consequences for the formation of collapsed structures in the universe, as has
been studied in the context of primordial black holes. We show that these tails
also affect the very-large-scale structures, making heavy clusters like "El
Gordo", or large voids like the one associated with the cosmic microwave
background cold spot, more probable. We compute the halo mass function and
cluster abundance as a function of redshift in the presence of exponential
tails. We find that quantum diffusion generically enlarges the number of heavy
clusters and depletes subhalos, an effect that cannot be captured by the famed
${f}_{\mathrm{NL}}$ corrections. These late-universe signatures could thus be
fingerprints of quantum dynamics during inflation that should be incorporated
in N-body simulations and checked against astrophysical data.