U.S. Citizenship and Immigration Services (USCIS) can deny a citizenship application if a person is in a polygamous or bigamous marriage. When applying, foreign nationals will be required to disclose on their Form N-400 whether they have ever been in a non-traditional marriage.
U.S. immigration law frowns upon being married to more than one person at a time. This typically can prohibit both bigamists and polygamists from being naturalized United States citizens.