While it is unclear whether the constant changes have led to cruel and unusual punishment, the United States Court of Appeals for the Ninth Circuit became so disturbed in 2012 about the expired drugs that it chastised the state, saying Arizona “has insisted on amending its execution protocol on an ad hoc basis.” While the court permitted the two executions to proceed and they went off without a hitch, the Ninth Circuit nonetheless observed that Arizona had a “rolling protocol that forces us to engage with serious constitutional questions and complicated factual issues in the waning hours before executions.”

Douglas A. Berman, an expert on criminal sentencing at Ohio State University, said corrections officials tended to have a cavalier attitude that might now be backfiring on them. As Mr. Berman archly put it, “What’s the big deal, as long as the guy ends up dead and I’m not literally torturing the guy along the way?” Prison officials and execution teams, he said, “don’t see any adjustment that they are making as likely to cause unnecessary suffering or pain.”

Image Joseph R. Wood III Credit... Arizona Department of Corrections, via Associated Press

There are, however, signs that suggest otherwise. Mr. Wood, 55, gasped — seemingly for air — more than 600 times before he died on July 23; his execution is now the subject of an independent investigation commissioned by the state. In January in Oklahoma, Michael Lee Wilson, 38, said, “I feel my whole body burning” right after the drugs used in his execution — a mix meant to paralyze him, render him unconscious and stop his heart — began flowing through his veins. He died moments later.