Harder the Task, Sparser the Representation: Sparsity as a Learning Signature of Capability in LLMs

In this work, we investigate how the internal representations of Large Language Models (LLMs) change with inputs of increasing difficulty. Although sparsity changes may not be obvious for individual samples, dataset-level results reveal a consistent statistical trend: as task difficulty increases (e.g., harder questions, longer contexts, or more answer choices), the last hidden states of LLMs become systematically sparser. In short, \emph{the harder the task, the sparser the representation}. This sparsity--difficulty relation is observable across diverse models and domains, suggesting that harder inputs drive more concentrated activation patterns in the last hidden state. Through a series of controlled analyses and learning-dynamics experiments, we show that representational density is a learned property of data familiarity: models develop rich, distributed representations for mastered patterns, while unfamiliar inputs default to sparser activations. Our finding is also actionable. We illustrate this with Sparsity-Guided Curriculum In-Context Learning (SG-ICL), which uses sparsity to select few-shot demonstrations matched to the query's difficulty, outperforming standard CoT and Auto-CoT baselines on MATH-500. Our study provides new insights into how LLM representations reflect task difficulty and how this signal can be leveraged for inference.