Chain-of-thought (CoT) supervision makes computationally hard learning problems tractable by exposing intermediate steps that are easy to learn via next-token prediction. However, in practice, CoT datasets draw on heterogeneous sources with different reasoning styles, and substantial effort goes into curating supervision suitable for next-token training. What role does this curation play in enabling the computational advantage of CoT supervision? In this work, we take a first step toward understanding this question from a learning-theoretic perspective. We introduce several models of $heterogeneous$ $CoT$ $supervision$ from multiple sources, each providing correct traces with the same final answer but different intermediate steps. Assuming that fitting a next-token predictor to $homogeneous$ traces from any single source is computationally tractable, we ask when this suffices for efficient learning from heterogeneous supervision. We show $cryptographic$ $hardness$ of learning from $passively$ collected traces from multiple sources, even though learning from homogeneous CoTs is easy. This hardness can stem either from $hard$-$to$-$learn$ $contribution$ $biases$ determining which sources contribute traces on different parts of the domain, or from receiving an $unfiltered$ heterogeneous mixture (though a filtered subset of homogeneous traces is sufficient for efficient learning). On the positive side, inspired by iterative and adaptive fine-tuning of language models, we give a computationally efficient learning algorithm that $actively$ and $adaptively$ collects CoTs from multiple sources with a per-source CoT budget completely independent of the target error $\varepsilon$. We argue that our results illuminate data collection and curation practices central to making CoT supervision effective for learning.

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