Well, it would seem to violate the "full faith and credit" clause. Some political analysts have weighed in on this already; here's
a fairly accessible write-up that also mentions a bill in Arkansas that tried to impose term limits by leaving people off the state ballot if they'd already served two terms.
Here's the full faith and credit discussion specifically:
Arizona’s bill, if it becomes law, would also seem to be vulnerable to a challenge under the Full Faith And Credit Clause. Section 1 of Article IV of the Constitution requires states to give full faith and credit to the public Acts, Records, and judicial Proceedings of every other State. This includes accepting as genuine records from a sister state that have been officially certified under seal from the appropriate record keeper. Under Arizona’s law, the Hawaii Certification Of Live Birth, which is an official document from the State of Hawaii, and the only birth record that the state releases. By failing to accept this document, even for the limited purpose that this law is written for, Arizona would be failing to give full faith and credit to the records of not just Hawaii, but every other state that only issues COLB’s as birth records.