Quantization has shown strong results in preserving model quality under compression. However, under aggressive bit-width reductions, even quantization may require additional information to prevent performance degradation. A natural source of it is second-order curvature information, captured by the Hessian. Since the Hessian of the model layers is prohibitively large, direct computation is infeasible, making structured parameterizations and approximations crucial in practice.In this work, we propose efficient Kronecker-factored approximation yielding state-of-the-art performance when integrated into existing quantization schemes. Evaluations on the LLaMA and Qwen model families show near-baseline quality at 4-bit compression and only a 5–6% degradation at 2-bit. Moreover, our method substantially accelerates the most expensive component in second-order quantization – Hessian parameterization – achieving up to a 10× speedup over prior approaches.