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One approach to reducing the massive costs of large language models (LLMs) is the use of quantized or sparse representations for training or deployment.While post-training compression methods are very popular, the question of obtaining even more accurate compressed models by *directly training* over such representations, i.e., *Quantization-Aware Training (QAT)*, is still open: for example, a recent study put the "optimal" bit-width at which models can be trained using QAT, while staying accuracy-competitive with standard FP16/BF16 precision, at 8-bits weights and activations. We advance this state-of-the-art via a new method called QuEST, for which we demonstrate optimality at 4-bits and stable convergence as low as 1-bit weights and activations. QuEST achieves this by improving two key aspects of QAT methods: (1) accurate and fast quantization of the (continuous) distributions of weights and activations via Hadamard normalization and MSE-optimal fitting; (2) a new *trust gradient estimator* based on the idea of explicitly minimizing the error between the noisy gradient computed over quantized states and the "true" (but unknown) full-precision gradient. Experiments on Llama-type architectures show that QuEST induces stable scaling laws across the entire range of hardware-supported precisions, and can be extended to sparse representations. We provide GPU kernel support showing that models produced by QuEST can be executed efficiently. Our code is available at [https://github.com/IST-DASLab/QuEST](https://github.com/IST-DASLab/QuEST).