We used atomistic simulations to study the origin of the change of resistance over time in the amorphous phase of GeTe, a prototypical phase-change material (PCM). Understanding the cause of resistance drift is one of the biggest challenges to improve multilevel storage technology. For this purpose, we generated amorphous structures via classical molecular-dynamics simulations under conditions as close as possible to the experimental operating ones of such memory devices. Moreover, we used the replica-exchange technique to generate structures comparable with those obtained in the experiment after long annealing that show an increase of resistance. This framework allowed us to overcome the main limitation of previous simulations, based on density-functional theory, that suffered from being computationally too expensive therefore limited to the nanosecond time scale. We found that resistance drift is caused by consumption of Ge atom clusters in which the coordination of at least one Ge atom differs from that of the crystalline phase and by removal of stretched bonds in the amorphous network, leading to a shift of the Fermi level towards the middle of the band gap. These results show that one route to design better memory devices based on current chalcogenide alloys is to reduce the resistance drift by increasing the rigidity of the amorphous network.