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Simulations of interacting electrons and bosons out of equilibrium, startingfrom first principles and aiming at realistic multiscale scenarios, is a grandtheoretical challenge. Here, using the formalism of nonequilibrium Green'sfunctions and relying in a crucial way on the recently discovered time-linearformulation of the Kadanoff-Baym equations, we present a versatile toolbox forthe simulation of correlated electron-boson dynamics. A large class of methodsare available, from the Ehrenfest to the dressed $GD$ for the treatment ofelectron-boson interactions in combination with perturbative, i.e.,Hartree-Fock and second-Born, or nonperturbative, i.e., $GW$ and $T$-matriceseither without or with exchange effects, for the treatment of the Coulombinteraction. In all cases the numerical scaling is linear in time and theequations of motion satisfy all fundamental conservation laws.

Time-linear scaling NEGF methods for real-time simulations of interacting electrons and bosons. I. Formalism