We study interaction-induced correlations in Luttinger liquid with multiple Fermi edges. Many-particle correlation functions are expressed in terms of Fredholm determinants ${\rm det}(1+\hat{A}\hat{B})$, where $A(\epsilon)$ and $B(t)$ have multiple discontinuities in energy and time spaces. We propose a general asymptotic formula for this class of determinants and provide analytical and numerical support to this conjecture. This allows us to establish non-equilibrium Fermi-edge singularities of many-particle correlation functions. As an example, we calculate a two-particle distribution function characterizing genuinely non-equilibrium quantum correlations between left- and right-moving fermions that have left the interaction region.