Like Jeff said, input is ignored from the frame that you press 1P start until the frame that you get control of Jumpman, so the RNG is locked. This makes manipulation of a savestate possible because you can start the board with an input held down and be frame-perfect every time, thus getting identical output.
You can test this yourself. Make two save states, one at the monkey climb screen, and one at the stacked monkey screen (or at any point between pressing 1P start and getting control of Jumpman).
Load the first state. Put your finger on the "left" key and hold it down. When you get control, you'll run behind the oil can. Keep your finger on left. Now watch the fireball/barrels.
Load the second state, keeping your finger on left (again, making sure it's before you get control of Jumpman so that the input will be frame-perfect). Watch the fireball/barrels. They'll do the exact same thing they just did.
You could also try loading the first state, holding down right, watching what happens, then loading the second state and holding right. You'll get different outputs than when you held left, but that second set of outputs will always be the same when you hold right.
Obviously you can't get frame-perfect other than for having the input held down initially, but that's still plenty exploitable because the fireball's first decisions (plus wild barrels, etc.) would all be pre-determined until your input deviates. Seems like it would be minor, but as we all know, 1-1 is all about taking advantage of "minor" things.
So it should be totally out of the question to use a savestate that is made after pressing P1 start.
You must watch the monkey climb!