**Version History**:

ArXiv v1, 24 April 2017 https://arxiv.org/abs/1704.07309v1

ArXiv v2, 25 April 2017 https://arxiv.org/abs/1704.07309v2

ArXiv v3, 9 September 2017 https://arxiv.org/abs/1704.07309v3

ArXiv v4, 5 October 2017 https://arxiv.org/abs/1704.07309v4

We initiate the study of computational entropy in the quantum setting. We investigate to what extent the classical notions of computational entropy generalize to the quantum setting, and whether quantum analogues of classical theorems hold. Our main results are as follows. (1) The classical Leakage Chain Rule for pseudoentropy can be extended to the case that the leakage information is quantum (while the source remains classical). Specifically, if the source has pseudoentropy at least \(k\), then it has pseudoentropy at least \(kââ \) conditioned on an \(â \)-qubit leakage. (2) As an application of the Leakage Chain Rule, we construct the first quantum leakage-resilient stream-cipher in the bounded-quantum-storage model, assuming the existence of a quantum-secure pseudorandom generator. (3) We show that the general form of the classical Dense Model Theorem (interpreted as the equivalence between two definitions of pseudo-relative-min-entropy) does not extend to quantum states. Along the way, we develop quantum analogues of some classical techniques (e.g. the Leakage Simulation Lemma, which is proven by a Non-uniform Min-Max Theorem or Boosting). On the other hand, we also identify some classical techniques (e.g. Gap Amplification) that do not work in the quantum setting. Moreover, we introduce a variety of notions that combine quantum information and quantum complexity, and this raises several directions for future work.

%B Poster presention at QIP 2017 and oral presentation at QCrypt 2017 %G eng %U https://arxiv.org/abs/1704.07309