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Is it $\theta$ or $\varphi$ as usually depicted on the Bloch sphere? In other words, is it the angle projected on the $xy$-plane or is it the one on a plane that intersects the $z$-axis of the Bloch sphere? How could I have answered this myself?

bloch sphere

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Neither.

Phase estimation algorithm does not estimate a property of a qubit state (and the angles $\theta$ and $\varphi$ in your question are exactly that - a property of a given qubit state).

Rather, it estimates the eigenvalue of a given unitary $U$ that corresponds to the given eigenvector $|\psi\rangle$ - the angle $\alpha$ in the following equation:

$$U|\psi\rangle = e^{2i\pi\alpha}|\psi\rangle$$

Wikipedia article on phase estimation has more details on the algorithm.

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