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
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.