This paper discusses variable or covariate selection for high-dimensional quadratic Cox model. Although many variable selection methods have been developed for standard Cox model or high-dimensional standard Cox model, most of them cannot be directly applied since they cannot take into account the important and existing hierarchical model structure. For the problem, we present a penalized log partial likelihood-based approach and in particular, generalize the regularization algorithm under marginality principle (RAMP) proposed in Hao et al. (J Am Stat Assoc 2018;113:615–25) under the context of linear models. An extensive simulation study is conducted and suggests that the presented method works well in practical situations. It is then applied to an Alzheimer’s Disease study that motivated this investigation.