States (Arizona and Kansas) have passed laws requiring voters to show proof of citizenship in order to register to vote. Meantime, a long-standing federal law (the National Voter Registration Act, or "Motor Voter" law) says that the U.S. Election Assistance Commission has to make a universal nationwide voter registration form, and that states have to take it and like it.
Long story short: Arizona's law already made it to the Supreme Court last year, when the Court required Arizona to recognize the federal form. However, Justice Scalia also suggested that the states might be able to sue the EAC to force them to add proof-of-citizenship language to the forms.
How well did that work out? Not too well, if the Tenth Circuit's recent opinion is the last word.