Which ancient empire left the greatest legacy? Why?

The Roman Empire.

Explanation:

The Roman Empire was an ancient state covering the Mediterranean basin, resulting from the transformation of the Roman republic into a monarchical system. It is assumed that the beginning of the empire was in 27 BCE, when Gaius Octavius received the title of Augustus (exalted by the gods) from the Senate. It confirmed Octavian's position as the most important person in the state and brought the final end to the Roman republic.  

Initially, the form of rule in the empire was the principate: the previous system was formally retained, and the emperor maintained the highest power in the state, exercising all the most important offices of the republic and controlling the army. As a result of the reforms of Diocletian and Constantine, the era of dominance came with a much stronger imperial position.  

In 395 AD, the empire finally broke up into the western and eastern parts. The Western Roman Empire, after the two barbarians conquered Rome, fell in 476 as a result of internal problems and the migration of peoples.

The Roman Empire is the predecessor, together with the Greek civilization, of the current Western culture: the European nations, to a greater or lesser extent, have received Roman influence through the territorial conquests that the Empire made, with which the great majority of nations have had cultural, political or idiomatic influences from Rome throughout its history.

i would say that the politicial parties from state to state varies.

independent innovation during the abbasid caliphate