Mochon's proof [Quantum Weak Coin Flipping with Arbitrarily Small Bias, preprint, arXiv:0711.4114, 2007] of the existence of quantum weak coin flipping with arbitrarily small bias is a fundamental result in quantum cryptography, but at the same time one of the least understood. Though used several times as a black box in important follow-up results [M. Ganz, Quantum Leader Election, preprint, arXiv:0910.4952, 2009; A. Chailloux and I. Kerenidis, in Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2009, pp. 527--533; N. Aharon and J. Silman, New J. Phys., 12 (2010), 033027; A. Chailloux and I. Kerenidis, in Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2011; I. Kerenidis and S. Zhang, in Theory of Quantum Computation, Communication, and Cryptography, Lecture Notes in Computer Science 7582, Springer, Berlin, 2013, pp. 13--28], the result has not been peer reviewed, its novel techniques (and, in particular, Kitaev's point game formalism) have not been applied anywhere else, and an explicit protocol is missing. We believe that truly understanding the existence proof and the novel techniques it relies on would constitute a major step in quantum information theory, leading to deeper understanding of entanglement and of quantum protocols in general. In this work, we make a first step in this direction. We simplify parts of Mochon's construction considerably, making about $20$ pages of analysis in the original proof superfluous, clarifying some other parts of the proof on the way, and presenting the proof in a way which is conceptually easier to grasp. We believe the resulting proof of existence is easier to understand, more readable, and certainly verifiable. Moreover, we analyze the resources neededto achieve a bias $\varepsilon$ and show that the number of qubits is $O(\log \frac{1}{\varepsilon})$, while the number of rounds is $(\frac{1}{\varepsilon})^{O(\frac{1}{\varepsilon})}$. A true understanding of the proof, including Kitaev's point-game techniques and their applicability, as well as completing the task of constructing an explicit (and also simpler and more efficient) protocol, are left to future work.

The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.

We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the GRAPH-ISOMORPHISM problem. We show that for the related problem of INDEX-ERASURE our method leads to a lower bound of square root of N which matches an upper bound obtained via reduction to quantum search on N elements. This closes an open problem first raised by Shi [FOCS'02]. Our approach is based on two ideas: (i) on the one hand we generalize the known additive and multiplicative adversary methods to the case of quantum state generation, (ii) on the other hand we show how the symmetries of the underlying problem can be leveraged for the design of optimal adversary matrices and dramatically simplify the computation of adversary bounds. Taken together, these two ideas give the new result for INDEX-ERASURE by using the representation theory of the symmetric group. Also, the method can lead to lower bounds even for small success probability, contrary to the standard adversary method. Furthermore, we answer an open question due to Spalek [CCC'08] by showing that the multiplicative version of the adversary method is stronger than the additive one for any problem. Finally, we prove that the multiplicative bound satisfies a strong direct product theorem, extending a result by Spalek to quantum state generation problems.