The Double Jeopardy Clause generally bars the government from prosecuting a person twice for the same offense, regardless of whether the first prosecution led to an acquittal or a conviction. (If the first prosecution led to a hung jury, reprosecution is allowed.) But the Court has long held that this only bars reprosecution by the same sovereign: The federal government may reprosecute a person after a state prosecution (and vice versa, and in principle the same for multiple states, in the rare cases where multiple states both have jurisdiction over the crime).
In Commonwealth v. Sanchez (2016), Justices Ginsburg, joined by Justice Thomas, argued that this doctrine should be reexamined:
I write only to flag a larger question that bears fresh examination in an appropriate case. The double jeopardy proscription is intended to shield individuals from the harassment of multiple prosecutions for the same misconduct. Current "separate sovereigns" doctrine hardly serves that objective. States and Nation are "kindred systems," yet "parts of ONE WHOLE." The Federalist No. 82. Within that whole is it not "an affront to human dignity," Abbate v. United States (1959) (Black, J., dissenting), "inconsistent with the spirit of [our] Bill of Rights," to try or punish a person twice for the same offense? Several jurists and commentators have suggested that the question should be answered with a resounding yes: Ordinarily, a final judgment in a criminal case, just as a final judgment in a civil case, should preclude renewal of the fray anyplace in the Nation. The matter warrants attention in a future case in which a defendant faces successive prosecutions by parts of the whole USA.
Today, the Court agreed to consider the issue, in Gamble v. U.S. For an earlier petition that makes what strikes me as a powerful originalist argument against the "separate sovereigns" doctrine, see this 2013 certiorari petition written by my UCLA colleague (and legal historian) Stuart Banner.