  We study the quantitative simplicity of the Lyapunov spectrum of
$d$-dimensional bounded matrix cocycles subjected to additive random
perturbations. In dimensions 2 and 3, we establish explicit lower bounds on the
gaps between consecutive Lyapunov exponents of the perturbed cocycle, depending
only on the scale of the perturbation. In arbitrary dimensions, we show
existence of a universal lower bound on these gaps. A novelty of this work is
that the bounds provided are uniform over all choices of the original sequence
of matrices. Furthermore, we make no stationarity assumptions on this sequence.
Hence, our results apply to random and sequential dynamical systems alike.
