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The Palmetto State could prove to be fertile political ground for Mr. Cruz, who bills himself as a defender of the Constitution, touts his pro-life credentials and is closely tied to former South Carolina Sen. Jim DeMint, who vacated his seat this year to take over the conservative think tank the Heritage Foundation.

A Princeton debate champion, Harvard Law School graduate, and former clerk for Chief Justice William H. Rehnquist, Mr. Cruz has earned headlines for the combative style he has brought to Capitol Hill since the 2012 election.

He opposed the nominations of Sen. John Kerry to be the next secretary of State and former Sen. Chuck Hagel to head the Pentagon. And he pushed back against Hurricane Sandy emergency relief funding for the Northeast, the Republican-controlled House’s plan to waive the nation’s borrowing limit for nearly four months and post-Newtown efforts to tighten the nation’s gun control laws.