What was the rivalry for world leadership between the us and the soviet union called?

The rivalry for world leadership between the United States and the Soviet Union is called the Cold War. The Cold War was a state of great political tension after World War II between the Eastern and Western Powers. Each thought the other would attack even though they never did. The threat of Nuclear warfare was a big idea that was rapidly spreading and it did not help the tension as Soviets continued to try and spread communism. 

The tried to make the government smaller by making the government smaller