That depends on which communist government you are asking about, which you don't say and the period of time.
no
The communist leader Mao Zedong became chairman of the Central People's Government of China in 1949. He held this position until 1959.
Nikita Khrushchev became head of government after Malenkov. Khrushchev already was head of the Communist Party, but consolidated his power by taking titular head of the Soviet Government as well.
krushchev
Senator Joe McCarthy was the leader of the anti Communist crusade of the 1950s in the US.
Communist states have had various names for their leaders. The leader of a communist state is usually called Chairman, as in the Chairman of the government. He was usually also the General Secretary of the Communist Party, which is the leader of the party. In communist states, the leader of the party was usually also the leader of the government. Leaders of communist countries often hold multiple positions. Today only Cuba, Laos, Vietnam, North Korea, and China are officially led by communist parties. The leader of China, Hu Jintao, is usually called President, but he is also the General Secretary of the Communist Party of China and the Chairman of the Central Military Commission of the People's Republic of China.
Vladimir Ilyich Lenin took the ideals from the Communist Manifesto and became a Russian revolutionary leader in the early 1900s.
Senator Joseph McCarthy was the leader of the anti-Communist crusade of the 1950s in the US.
China's 'Communist Revolution' was the Chinese Civil War. The leader of the Communist group was Mao Zedong. The leader of the standing, nationalist government was Chiang Kai-Shek.
Mikhail Gorbachev became the General Secretary of the Communist Party of the Soviet Union, and thus the leader of the USSR, in 1985.
Mao Zedong
Russia had a major revolution in 1917 that started the communist government and the name of the country became the USSR (Union of Soviet Socialist Republics) or the Soviet Union. The czar Nicolas II was executed and Vladmir Lenin became the leader.