Minnesota has a criminal law that punishes complaints of police misconduct that a person knows to be false. The constitutionality of that law has been challenged and may now be on its way to the U.S. Supreme Court.
Here's some background. Melissa Crawley complained that an officer forged her signature on a medical release form at a hospital. Police looked into the matter and concluded that Crawley fed them false information. Crawley was charged under the Minnesota law and was later convicted by a jury.