With an increasing number of antennas in multiple-input multiple-output (MIMO) systems, the energy consumption and costs of the corresponding front ends become relevant. In this context, a promising approach is the consideration of low-resolution data converters. In this study two novel optimal precoding branch-and-bound algorithms constrained to constant envelope signals and phase quantization are proposed. The first maximizes the minimum distance to the decision threshold (MMDDT) at the receivers, while the second minimizes the MSE between the users data symbols and the receive signal. This MMDDT design presented in this study is a generalization of prior designs that rely on 1-bit quantization. Moreover, unlike the prior MMSE design that relies on 1-bit resolution, the proposed MMSE approach employs uniform phase quantization and the bounding step in the branch-and-bound method is different in terms of considering the most restrictive relaxation of the nonconvex problem, which is then utilized for a suboptimal design also. Moreover, three different soft detection methods and an iterative detection and decoding scheme that allow the utilization of channel coding in conjunction with low-resolution precoding are proposed. Besides an exact approach for computing the extrinsic information, two approximations with reduced computational complexity are devised. The proposed branch-and-bound precoding algorithms are superior to the existing methods in terms of bit error rate. Numerical results show that the proposed approaches have significantly lower complexity than exhaustive search. Finally, results based on an LDPC block code indicate that the proposed receive processing schemes yield a lower bit-error-rate compared to the conventional design.