Large language models (LLMs) incur prohibitive compute and energy costs, and applying sparsity, quantization, or low‑rank approximation alone often hits an accuracy‑efficiency wall.\ \ This work introduces the Compression Trinity framework, which jointly employs sparsity to cut arithmetic, quantization to shrink memory bandwidth, and low‑rank approximations to recover the accuracy lost by the other two.\ \ During pre‑training, MKOR approximates curvature with block‑diagonal sparsity and low‑rank inversion, preserving numerical stability for quantized states and reducing curvature update complexity from $O(d^3)$ to $O(d^2)$. Compared with KFAC, convergence speeds up by about 1.85×.\ \ SLoPe accelerates training by up to 1.25× through a double‑pruned backward pass for N:M sparsity, and in the final 1% of training it inserts low‑rank “lazy” adapters to regain accuracy lost to sparsity.\ \ For post‑training compression, OPTIMA formulates weight reconstruction as a globally optimal column‑wise quadratic program, stabilizing static masks in a zero‑training regime and improving zero‑shot accuracy by roughly 3.97%. Given a fine‑tuning budget, PATCH learns a dynamic hybrid sparsity ratio between 0% and 50%, breaking the static‑mask ceiling and achieving up to 1.38× speedup.\ \ SLiM realizes the full Trinity in a single shot, using mathematically derived low‑rank adapters to compensate for information lost to quantization and sparsity, boosting accuracy by about 5.66% and surpassing uncompressed dense models by 0.6% at equal parameter budgets.\ \ Overall, the joint application of sparsity, quantization, and low‑rank approximation is essential for building efficient, scalable LLMs.\ \ Blogger's Review: The Trinity perspective turns compression tricks into a coherent system rather than isolated patches, and the empirical gains across pre‑training and post‑training stages make a compelling case for broader adoption in real‑world deployments.