The early time dynamics in systems of highly non-equilibrium confined carriers incorporates a number of interesting quantum phenomena. Such systems are carriers, injected or optically excited in a nanowire, which interact with optical phonons. The generalized Wigner function provides a convenient approach for investigation of the ultrafast electron-phonon kinetics. Two quantum-kinetic models are derived which can be regarded as counterparts of the Levinson (L) and the Barker-Ferry (B-F) equations, now generalized to account for a space-dependent evolution in nanowires. A comparison with the classical Boltzmann kinetics reveals the basic peculiarities of the quantum evolution: the electron-phonon interaction has a finite duration, particular collisions do not conserve energy, the evolution is non-Markovian. The models differ in the way collisions with different duration times are treated: the effects of the events with a long duration are suppressed in the B-F model. The choice of III-V materials at very low temperatures provides a clear reference picture, where classical electrons can only emit phonons, since the constant polar phonon energy forms replicas of the initial distribution. The energy region above the initial condition is forbidden: the fastest carriers are the ballistic electrons. The non-Markovian evolution gives rise to a retardation in the build-up of the replicas, which is larger in the B-F model. The lack of energy conservation causes a broadening of the replicas and the appearance of electrons in the classically forbidden energy region. The effect of ultrafast spatial transfer is observed. Certain quantum carriers reach larger distances than the classically fastest ballistic electrons. The modification of the classical trajectory due to the finite collision process has an important physical effect. If neglected, it leads to negative densities around the front of the fastest quantum electrons.