Are states generally free to decide on term caps concerning the election of senators and members of Congress?
If a state wants to enforce a four term cap on congress people, for instance, is that law simply one amendment of a state constitution away? What about if a state would like to have a born-in-the-USA law concerning senators like what exists for the presidency?
Would all such decisions be made at a state level, or would this be decided at the national/federal level?
See U.S. Term Limits, Inc. v. Thornton, 514 U.S. 779 (1995):
Today's cases present a challenge to an amendment to the Arkansas State Constitution that prohibits the name of an otherwise-eligible candidate for Congress from appearing on the general election ballot if that candidate has already served three terms in the House of Representatives or two terms in the Senate. The Arkansas Supreme Court held that the amendment violates the Federal Constitution. We agree with that holding. Such a state-imposed restriction is contrary to the "fundamental principle of our representative democracy," embodied in the Constitution, that "the people should choose whom they please to govern them." Allowing individual States to adopt their own qualifications for congressional service would be inconsistent with the Framers' vision of a uniform National Legislature representing the people of the United States. If the qualifications set forth in the text of the Constitution are to be changed, that text must be amended.
Jen's answer regarding U.S. Term Limits v. Thornton is correct. This extends beyond just term limits, though. In general, courts have held that, for offices for which the U.S. Constitution establishes eligibility criteria, neither states nor Congress can add further ones (aside from by amending the U.S. Constitution accordingly.) This applies to elections for members of Congress as well as those for President and Vice President, since the Constitution establishes specific eligibility requirements for those offices.
While the question was only tested specifically on term limits for Congress in 1995, this precedent has held for essentially the entire history of the U.S. Powell v. McCormack held in 1969 that the House erred when it refused to seat a newly-elected member due to a state law holding that person ineligible, ruling that the state law was unconstitutional and that the House could not refuse to seat the member. Prior to their attempt to do so in 1969, it had been the established practice of Congress all the way back to at least 1807 to ignore any state-imposed eligibility requirements when seating members, since those were deemed to be inherently unconstitutional.
