What was one reason for the decline of both the han dynasty and the western roman empire?

 The Han fell during the what is known as the Three Kingdoms period. The 3 kingdoms period is characterized best as a fractious empire ruled by local warlords and little central oversight. You might remember that the years of decline had many divisions of the empire such as Diocletian with his invention of having 2 Caesars and 2 Augusti known as the Tetrarchy. The fall of the Han and the fall of the Roman empire can each be characterized as being preceded by a time of division. One of the most important books in all of Chinese literature is entitled Romance of the Three Kingdoms and it begins with the phrase: "The Empire, long divided, must unite: long united, must divide." So like the Han Empire, the Roman Empire can be characterized by periods of division and reunification over several hundred years before its ultimate demise. 

The big difference between the Han and the Romans is that the Hans fell from within, they were nor conquered by an outside force as was the case with the Romans. This can be contrasted with the Yuan dynasty of China which WAS an outside "barbarian" force (the mongols). Furthermore, the culture of the Han did not fundamentally change with its fall with one exception, the introduction of Buddhism. Like the introduction of Christianity in Rome, the Chinese saw the introduction of a major religion that would endure to this day. You should note that China did not have a unified religion but a unified culture whereas Rome demanded a unified, state religion. China tolerated differing philosophies such as Taoism Buddhism and Confucianism, whereas pagans and Christians (later Orthodox and Arians) had significant struggles. The pagans blamed the Christians for the sack of Rome in the 5th century which led to St. Augustine to publish his work City of God. Barbarian kings such as Odoacer and Theodoric brought Arian Christians into Orthodox Italy. The Roman capital of Ravenna has both Arian and Orthodox churches and baptisteries which were constructed during this period. The cultural diversity during this period is significant compared to China during the fall of the Han and the rise of the Jin. 

The "riot" you are looking for during the Han dynasty is the "Yellow Turban Rebellion" which set off a series of dramatic results that ultimately destroyed the empire. The book Romance of the Three Kingdoms begins with the Yellow Turban Rebellion. This revolt provided an opportunity for many young statesmen in their 20s and 30s to gain valuable leadership and military experience. When the Han fractured, these warlords began fighting for domination with three rulers emerging victorious: Cao ruler of Wei, Liu ruler of Shu, and Sun ruler of Wu. Each participated in quelling the rebellion to restore the Han in their early years but later attempted to assume power themselves. The ultimate victor was a sub general of Wei known as Sima Yi who staged a coup from the Cao family and unified all of China. The ultimate victor over Rome were the Germans.

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