Recent progress in computer vision-oriented neural network designs is mostly driven by capturing high-order neural interactions among inputs and features. And there emerged a variety of approaches to accomplish this, such as Transformers and its variants. However, these interactions generate a large amount of intermediate state and/or strong data dependency, leading to considerable memory consumption and computing cost, and therefore compromising the overall runtime performance. To address this challenge, we rethink the high-order interactive neural network design with a quadratic computing approach. Specifically, we propose QuadraNet -- a comprehensive model design methodology from neuron reconstruction to structural block and eventually to the overall neural network implementation. Leveraging quadratic neurons' intrinsic high-order advantages and dedicated computation optimization schemes, QuadraNet could effectively achieve optimal cognition and computation performance. Incorporating state-of-the-art hardware-aware neural architecture search and system integration techniques, QuadraNet could also be well generalized in different hardware constraint settings and deployment scenarios. The experiment shows thatQuadraNet achieves up to 1.5$\times$ throughput, 30% less memory footprint, and similar cognition performance, compared with the state-of-the-art high-order approaches.