Arizona Republicans might introduce a bill this fall that would deny birth certificates to children born in the United States, but whose parents are not lawfully present in the country. Presumably, the purpose of the law is to make it difficult or impossible for these children to establish their citizenship.
The Fourteenth Amendment provides that "all persons, born. . .in the United States, and subject to the jurisdiction thereof, are citizens of the United States" (italics added). This is known as "birthright citizenship." Some Republicans have sought ways to eliminate birthright citizenship, including by constitutional amendment.
This constitutional provision reverses the infamous Supreme Court decision in Dred Scott v. Sanford, which held that blacks -- whether slave or free -- were not citizens of the United States and therefore were not entitled to assert rights in US courts. Apparently, some lawmakers in Arizona want to bring back a portion of the law stated in Dred Scott.
I suspect that most of the conservatives on the Court would rule that this law violates the Equal Protection Clause. Although the Constitution is ambiguous in most places, this language is pretty clear. Furthermore, while the Constitution does not mandate that states issue birth certificates, it does not allow states to treat citizens as if they were aliens.