What I believe to be several insurmountable problems;
1. If any state should decide to require that a candidate prove his citizenship the simple presentation of a certified birth certificate would be all that is required since this is the primary document accepted by the United States government to establish citizenship.
2. In order to challenge the authenticity of the birth certificate a state's secretary of state would have to demonstrate that the said document is a forgery, the only problem there is that if the birth certificate is certified by the custodian of records of the issuing document and that certification is verified by the issuing state the challenging secretary of state will be left no choice but to accepted under the Full Faith and Credit Clause of the Constitution.
3. If a candidate presents a certified copy of their birth certificate, I doubt that a state could impose a greater burden since the states do not have the right or the power to go beyond the requirements laid out under the Constitution, and as I pointed out in point #1, a certified birth certificate is prima facia evidence of citizenship and higher burden of proof by a state would in all likelihood be deemed unconstitutional.
4. The only persons who would be able to challenge a candidates eligibility in federal court would be a challenging secretary of state or another candidate who can also show that they would be harmed by the challenged candidates appearance on the ballot. All others would lack standing.
5. I would think that it would be highly unlikely for the Supreme Court, or any lower federal court to here a case requiring a definitive answer to the question of a candidates eligibility considering the fact that this is a political process in which challenges would be properly raised by the Electoral College and the subsequent joint secession of the Congress based upon the "
Political Question Doctrine".
Finally, If such a case were ever to be heard before a federal court it would the result of a candidate being barred from the ballot. In this case I seriously doubt that the plaintiff would lose for the reason expressed above. In short all the arguments about Vattel, the definition of "natural born" etc., while fun and entertaining, are moot points.