I am going through Ryan O'Donnell's quantum computing class homeworks, and on problem 6 of this assignment you are asked to prove $QMA$ in $PP$ using the fact that $RestartingPP = PP$, where $RestartingPP$ contains the languages $L$ for which there exists a "restarting PP machine" that decides it in the usual $PP$ sense, where, for a general randomized model of computation $M$, a "restarting $M$ machine" is defined as follows:
We define [a restarting $M$ machine] as follows: At any time during the $M$-computation, the machine is allowed to “hit the restart button” (if you like, think of this as entering a special kind of Turing Machine state). When this occurs, every aspect of the computation is set back to its initial value, and computation begins again. Nothing from the earlier computation is remembered. The machine may do multiple restarts, and each is called a run. Assuming that we are interested in the time complexity of M, we define the time complexity of a Restarting $M$ computation to be the maximum of the time of the runs (not the total time). Basically, if the $M$ computation does not like the way a computation is going, it can freely restart (but has to forget everything).
Now, I get that this is essentially what is going on with post selection (Kuperberg even calls post selection the "free to retry" property (link)). What confuses me, though, is that to prove $QMA \subseteq PP$, one typically uses strong error reduction for $QMA$ (as in Marriott-Watrous amplification (link)), and here, in this homework set, it is implicitly implied that you actually don't need strong error reduction for this result. I'd really appreciate it if someone could walk me through this proof using the restarting machine approach, because I don't see how to do it without using strong error reduction and the usual technique described in this TCS stack exchange post.
