  Reconstructing a signal from squared linear (rank-one quadratic) measurements
is a challenging problem with important applications in optics and imaging,
where it is known as phase retrieval. This paper proposes two new phase
retrieval algorithms based on non-convex quadratically constrained quadratic
programming (QCQP) formulations, and a recently proposed approximation
technique dubbed feasible point pursuit (FPP). The first is designed for
uniformly distributed bounded measurement errors, such as those arising from
high-rate quantization (B-FPP). The second is designed for Gaussian measurement
errors, using a least squares criterion (LS-FPP). Their performance is measured
against state-of-the-art algorithms and the Cram\'er-Rao bound (CRB), which is
also derived here. Simulations show that LS-FPP outperforms the state-of-art
and operates close to the CRB. Compact CRB expressions, properties, and
insights are obtained by explicitly computing the CRB in various special cases
-- including when the signal of interest admits a sparse parametrization, using
harmonic retrieval as an example.
