I'd have to refresh my memory on this, but I think there was some discussion during the Constitutional debates about there being a joint presidency. I know I've read several alternate-history stories where there was a dual presidency. I think there's some symmetry to an Executive Branch that has a council of nine co-presidents, like the nine justices of the Judicial Branch. There are a few pluses and minuses that I can think of off the top of my head. There would be no need for a vice president, for one thing. They should not be all on the same ticket and elected separately, which would be good-- but could also be bad, if any of them decided they were only representing their part of the country, as Congress Critters do. Most of all, I wonder how limber a government could be with multiple heads of state. Of course, you could always structure it to take advantage of the inevitable: Say, five co-presidents with one Chief President and then Presidents of the North, South, East, and West. But that kind of makes them sound like wicked witches more than presidents.
How about a streamlined government of five presidents as above, one senator from each state, no house, and nine justices?