We study quantized beamforming in wireless amplify-and-forward relay-interference networks with any number of transmitters, relays, and receivers. We design the quantizer of the channel state information to minimize the probability that at least one receiver incorrectly decodes its desired symbol(s). 

For networks with a single transmitter-receiver pair, we provide necessary and sufficient conditions on the structure of optimal quantizers. In particular, we show that it is necessary and sufficient to use relay selection to achieve maximal diversity. We also prove that the average SNR loss and the capacity loss due to quantization decays at least exponentially with the number of feedback bits. 

For multiuser networks, we introduce a generalized diversity measure that encapsulates the conventional one as the \textit{first-order} diversity. Additionally, it incorporates the \textit{second-order} diversity, which is concerned with the transmitter power dependent logarithmic terms that appear in the error rate expression. First, we show that, regardless of the quantizer and the amount of feedback that is used, the relay-interference network suffers a second-order diversity loss compared to interference-free networks. Then, two different quantization schemes are studied: First, using a global quantizer, we show that a simple relay selection scheme can achieve maximal diversity. Then, using the localization method, we construct both fixed-length and variable-length local (distributed) quantizers (fLQs and vLQs). Our fLQs achieve maximal first-order diversity, whereas our vLQs can achieve full diversity with arbitrarily low feedback rates when the transmitter powers are sufficiently large.