Where does Allaah instruct in Qur'an establishment of a caliphate?

BQ: Prophet Muhammad gave prophecy that Muslims would follow the footsteps of the prior religions and that Islam would splinter and be corrupted in various ways (following in the same errors as the religions prior to Islam). With that in mind, does it make sense to suggest authentic hadeeth that claim establishment of a caliphate is mandated by Allaah?