In phase-change memory devices, a material is cycled between glassy and crystalline states. The highly temperature-dependent kinetics of its crystallization process enables application in memory technology, but the transition has not been resolved on an atomic scale. Using femtosecond x-ray diffraction and ab initio computer simulations, we determined the time-dependent pair-correlation function of phase-change materials throughout the melt-quenching and crystallization process. We found a liquid-liquid phase transition in the phase-change materials Ag4In3Sb67Te26 and Ge15Sb85 at 660 and 610 kelvin, respectively. The transition is predominantly caused by the onset of Peierls distortions, the amplitude of which correlates with an increase of the apparent activation energy of diffusivity. This reveals a relationship between atomic structure and kinetics, enabling a systematic optimization of the memory-switching kinetics.