The was the state of political hostility between countries without an outright declaration of war.

The right answer is the Cold War.

The Cold War refers to the hostile relationship that, after World War II (1939-1945), and until 1980, existed primarily between the United States and the Soviet Union. These countries had fought as allies in World War II, but, even then, they had distrusted each other. Soon after the war ended, the distrust grew stronger, and both nations, exponents, respectively, of capitalism (U.S.) and communism (USSR), struggled to become the world's dominating power by undermining (through propaganda, espionage, and other means) their dissenting political regimes.  

The conflict owns its name to the fact that there was never a warfare between the two major powers, although other nations fought on their behalf (Vietnam and Afghanistan, among them).