Which branch of government has exercised the most power regarding race to the top, common core standards, and its assessments? which has exercised the least?

For the first question, the answer is the judicial branch of the government since it is the one who ordain and establishes. The Judicial branch construes the laws. The judiciary also offers a tool for the resolution of arguments.

For the second question, the answer is the executive branch of the government. The executive is the structure exercising authority in and holding accountability for the power of a state. The executive implements and imposes law.