  We analyse the long-lasting effects of initial conditions on fluctuations in
one-dimensional diffusive systems. We consider both the fluctuations of current
for non-interacting diffusive particles starting from a step-like initial
density profile, and the mean-square displacement of tracers in homogeneous
systems with single-file diffusion. For these two cases, we show analytically
(via the propagator and Macroscopic Fluctuation Theory, respectively) that the
long-term memory of initial conditions is mediated by a single static quantity:
a generalized compressibility that quantifies the density fluctuations of the
initial state. We thereby identify a universality class of hyperuniform initial
states whose dynamical variances coincide with the `quenched' cases studied
previously; we also describe a continuous family of other classes among which
equilibrated (or `annealed') initial conditions are but one family member. We
verify our predictions through extensive Monte Carlo simulations.
