  Low-resolution analog-to-digital converters (ADCs) have emerged as a
promising technology for reducing power consumption and complexity in massive
multiple-input multiple-output (MIMO) systems while maintaining satisfactory
spectral and energy efficiencies (SE/EE). In this work, we first identify the
essential properties of optimal quantization and leverage them to derive a
closed-form approximation of the covariance matrix of the quantization
distortion. The theoretical finding facilitates the system SE analysis in the
presence of low-resolution ADCs. We then focus on the joint optimization of the
transmit-receive beamforming and bit allocation to maximize the SE under
constraints on the transmit power and the total number of active ADC bits. To
solve the resulting mixed-integer problem, we first develop an efficient
beamforming design for fixed ADC resolutions. Then, we propose a low-complexity
heuristic algorithm to iteratively optimize the ADC resolutions and beamforming
matrices. Numerical results for a $64 \times 64$ MIMO system demonstrate that
the proposed design offers $6\%$ improvement in both SE and EE with $40\%$
fewer active ADC bits compared with the uniform bit allocation. Furthermore, we
numerically show that receiving more data streams with low-resolution ADCs can
achieve higher SE and EE compared to receiving fewer data streams with
high-resolution ADCs.
