Test-time compute scaling has demonstrated the ability to improve the performance of reasoning language models by generating longer chain-of-thought (CoT) sequences. However, this increase in performance comes with a significant increase in computational cost. In this work, we investigate two compute constraint strategies: (1) reasoning length constraint and (2) model quantization, as methods to reduce the compute demand of reasoning models and study their impact on their safety performance. Specifically, we explore two approaches to apply compute constraints to reasoning models: (1) fine-tuning reasoning models using a length controlled policy optimization (LCPO) based reinforcement learning method to satisfy a user-defined CoT reasoning length, and (2) applying quantization to maximize the generation of CoT sequences within a user-defined compute constraint. Furthermore, we study the trade-off between the computational efficiency and the safety of the model.