Assume $f(x)$ is n-bit to n-bit function. Let $F(x)$ be defined as $T$ iterations of $f(x)$, i.e. $F(x) = f^T(x)$.
Quantum algorithm relies on $F(x)$; it calls it $R$ times. What is the best query complexity of the algorithm in terms of calls to $f(x)$:
Can we do better than $R \cdot T$ queries while maintaining negligible quantum memory complexity?
Can we do better than $R \cdot T$ queries with additional quantum memory? If so, then how much? Can we do less than $T$?