A simpler proof of existence of quantum weak coin flipping with arbitrarily small bias

Mochon's proof [Moc07] of 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Though used several times as a black box in important follow-up results [Gan09, CK09, AS10, CK11, KZ13] the result has not been peer-reviewed, its novel techniques (and in particular Kitaev's point game formalism) have not been applied anywhere else, and an explicit protocol is missing.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 needed to achieve a bias $\epsilon$ and show that the number of qubits is $O(\log 1/\epsilon)$, while the number of rounds is $(1/\epsilon)^{O(1/\epsilon)}$. 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.