Hadamard Product (HP) has long been a cornerstone in click-through rate (CTR) prediction tasks due to its simplicity, effectiveness, and ability to capture feature interactions without additional parameters. However, the underlying reasons for its effectiveness remain unclear. In this paper, we revisit HP from the perspective of Quadratic Neural Networks (QNN), which leverage quadratic interaction terms to model complex feature relationships. We further reveal QNN's ability to expand the feature space and provide smooth nonlinear approximations without relying on activation functions. Meanwhile, we find that traditional post-activation does not further improve the performance of the QNN. Instead, mid-activation is a more suitable alternative. Through theoretical analysis and empirical evaluation of 25 QNN neuron formats, we identify a good-performing variant and make further enhancements on it. Specifically, we propose the Multi-Head Khatri-Rao Product as a superior alternative to HP and a Self-Ensemble Loss with dynamic ensemble capability within the same network to enhance computational efficiency and performance. Ultimately, we propose a novel neuron format, QNN-alpha, which is tailored for CTR prediction tasks. Experimental results show that QNN-alpha achieves new state-of-the-art performance on six public datasets while maintaining low inference latency, good scalability, and excellent compatibility. The code, running logs, and detailed hyperparameter configurations are available at: https://github.com/salmon1802/QNN.