which arguments were used by the Pope and the emperor in the Middle Ages to justify their superior power over another

Answer: popes: it was them who constitued legacy of Jesus Christ in the West and claimed for themselves a positon of God's ministers. Emperors: popes and church should follow Jesus Christ's example and the ecclesiastical doctrine. They (popes) should not be dominated by temporal/profane power and material goods.

Explanation: Political ambitions of the church were provable but also logical because who disposed human capital necessary for building effective society was a church (at least during early middle ages). Disputes between popes and emperors became increasingly frequent in the 14th century. But there are also many examples of their collaboration.