The influence by Greece was mainly by their golden age and Rome with its great Empire and Republic. Ancient Rome formed the law code much like the one used in the present time in many countries.
The ancient Roman republic had three branches of government. In the beginning, the legislative branch was the Senate, a group made up of 300 citizens from Rome's patrician class, the oldest and wealthiest families of Rome. It was the patricians, tired of obeying the king, who revolted and threw out Tarquinius Superbus. The Senate was the most powerful branch of the Roman republic, and senators held the position for life. The executive branch was made up of two consuls, elected yearly. These two consuls had almost kingly powers, and each could veto, or disapprove of the other's decision. It is quite possible that the idea of two consuls came from Sparta with its two kings. Praetors were part of the judicial branch, they were elected yearly by the people of Rome, and acted as judges.
The correct answer to this open question is the following.
The power of Congress that Senator Paul probably most concerned about based on this passage is the Power to Declare War.
The United States Constitution states on Article I, section clause 1, that the United States Congress has the power to declare war. The Constitution grants Congress with expressed powers and this is one of them. That is why Senator Rand Paul is reflecting on the great responsibility Congress has regarding thus power of declaring war against another nation.