This paper proposes a novel matrix quantization method, Binary Quadratic Quantization (BQQ). In contrast to conventional first-order quantization approaches, such as uniform quantization and binary coding quantization, that approximate real-valued matrices via linear combinations of binary bases, BQQ leverages the expressive power of binary quadratic expressions while maintaining an extremely compact data format. We validate our approach with two experiments: a matrix compression benchmark and post-training quantization (PTQ) on pretrained Vision Transformer-based models. Experimental results demonstrate that BQQ consistently achieves a superior trade-off between memory efficiency and reconstruction error than conventional methods for compressing diverse matrix data. It also delivers strong PTQ performance, even though we neither target state-of-the-art PTQ accuracy under tight memory constraints nor rely on PTQ-specific binary matrix optimization. For example, our proposed method outperforms the state-of-the-art PTQ method by up to 2.2\% and 59.1% on the ImageNet dataset under the calibration-based and data-free scenarios, respectively, with quantization equivalent to 2 bits. These findings highlight the surprising effectiveness of binary quadratic expressions for efficient matrix approximation and neural network compression.