Scaling up Large Language Models (LLMs) has significantly enhanced their capabilities but at the cost of increased resource demands during operation. As a result, compressing these large pre-trained models has become a preferred approach for creating smaller, resource-efficient customized versions. However, the effectiveness of compression as a method to maintain both efficiency and trustworthiness in LLMs is still uncertain, considering that compression usually does not perform alignment as pre-training. This study conducts an unprecedented, thorough evaluation of three (3) leading LLMs using five (5) state-of-the-art compression techniques across (8) trustworthiness dimensions, encompassing a range of compression rates. Our extensive experiments reveal the nuanced impacts of model compression on LLM trustworthiness. For example, while quantization (up to 4-bit) can maintain a level of trustworthiness comparable to that of uncompressed models, existing pruning approaches often struggle to sustain consistent reliability at 50\% sparsity. We also note that quantization with a moderate bit range can enhance specific trustworthiness aspects, like privacy and fairness. However, at very low bit levels (such as 3 bits), there is a marked decline in performance across various trust-related dimensions. These findings culminate in practical recommendations for balancing efficiency and trust in LLMs, providing invaluable insights for efficient and responsible AI deployment.