As speech processing systems in mobile and edge devices become more commonplace, the demand for unintrusive speech quality monitoring increases. Deep learning methods provide high-quality estimates of objective and subjective speech quality metrics. However, their significant computational requirements are often prohibitive on resource-constrained devices. To address this issue, we investigated binary activation maps (BAMs) for speech quality prediction on a convolutional architecture based on DNSMOS. We show that the binary activation model with quantization aware training matches the predictive performance of the baseline model. It further allows using other compression techniques. Combined with 8-bit weight quantization, our approach results in a 25-fold memory reduction during inference, while replacing almost all dot products with summations. Our findings show a path toward substantial resource savings by supporting mixed-precision binary multiplication in hard- and software.