I threw this in mostly as a hedge so that someone who knows better doesn't slap me around with the Hatch act.
It seems like the converse is true, though, states don't get to define qualifications - U.S. Term Limits, Inc. v. Thornton, 514 U.S. 779 (1995)
"Second, even if the States possessed some original power in this area, it must be concluded that the Framers intended the Constitution to be the exclusive source of qualifications for Members of Congress, and that the Framers thereby "divested" States of any power to add qualifications."
It seems like the converse is true, though, states don't get to define qualifications - U.S. Term Limits, Inc. v. Thornton, 514 U.S. 779 (1995)
"Second, even if the States possessed some original power in this area, it must be concluded that the Framers intended the Constitution to be the exclusive source of qualifications for Members of Congress, and that the Framers thereby "divested" States of any power to add qualifications."