What is the complexity of the quantum phase estimation in Grover's algorithm?

Suppose we are using GA (Grover’s algorithm) such that we are given it has 2 or more solutions. The search space is of size $N$. We all know Grover’s algorithm has, at worst, a time complexity proportional to $sqrt{N}$.
Now assume we are using quantum phase estimation to get the exact number of solutions in GA to precision $p$ bits (which is actually an angle on the complex unit circle) – so this means we will need to apply quantum phase estimation to get this angle – from this we get the exact numbers of solutions in GA. Using GA with phase estimation in this way is well known.

My question is, does the quantum phase estimation algorithm, in the worst case, run in polynomial time and polynomial space according to $N$?