The Post-Atomic Horror was a period in Earth history following the end of World War III in 2053, and running through the following decades. This period was marked by a breakdown in civil society and government and the denial of rights to alleged criminals.
Earth had emerged from this period by the early 22nd century, thanks to the assistance of the Vulcans, who made first contact with humanity during this period, in 2063. (TNG movie: Star Trek: First Contact; Star Trek: Myriad Universes novel: A Less Perfect Union)
It was believed by some latter era historians that the Post-Atomic Horror helped breed a more enlightened humanity, and was necessary for their eventual development into a responsible species. (TOS novel: Captain's Peril)