During his tenure as Permanent Representative to the United Nations, Alarcón held several prominent offices, such as President of the Security Council and Vice-President of the General Assembly.

[4] In 1962, the new Castro-led government appointed Alarcón as the Director of the Ministry of Foreign Affairs' Americas division, where he began his diplomatic career.

[5] Alarcón took over the office of President of the National Assembly of People's Power in 1993 in what Ben Corbett, a historian, considered a "demotion" from his earlier post as Minister of Foreign Affairs.

[8] One year after taking office, Alarcón travelled to the United States as the head of a five-member delegation to talk about the migration issues between the two countries.

Alarcón, along with the Cuban government, believed that the United States economic embargo against Cuba was the main culprit for mass emigration from the country.

[12] In August 2000, Alarcón was involved in a minor dispute with the United States when he was denied a visa to attend an international conference in New York City.

Alarcón lived in Manhattan for over twelve years, but because of his status as a Cuban government official, he was only allowed within a 25-mile radius of the United Nations.

[17] Alarcón, in an interview on the topic on who would succeed Fidel Castro, said; "All those who have been trying to fool the world and put out the idea that something terrible would happen in Cuba, that people would take to the streets, that there would be great instability, the door slammed on them and they must have very swollen hands now".