Phase-retrieval (PR) receivers can reconstruct complex-valued signals using only direct detection without the use of any optical carriers. We propose and demonstrate two PR receiver solutions with faster and better convergence. First, we demonstrate a PR receiver based on parallel alternative projections that are produced by propagating the signal through an array of dispersive elements of increasing length followed by direct detection. Fast convergence and high retrieved phase accuracy are achieved using a modified Gerchberg-Saxton (GS) algorithm that uses each projection as an intensity constraint. Second, we achieve similar performances employing an enhanced single projection GS algorithm with selective phase reset using symbol-wise GS errors. We experimentally reconstruct a 30 Gbaud QPSK signal after 55 km single-mode fiber transmission using the proposed solutions with a reduced number of iterations.