What legal authority has jurisdiction over a state supreme court that has acted improperly?
That depends on the nature of the improper act. If the state Supreme Court violated a federal law or right protected by the constitution, you can appeal to the U.S. Supreme Court. However, if you think they acted improperly in regards to a state law, then you are out of luck. Note: This answer assumes that by “state supreme court” you mean the highest court in the state. Some states (e.g.