2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/91947A family of protocols for quantum weak coin-flipping which asymptotically achieve a bias of 0.192 is described in this paper. The family contains protocols with n+2 messages for all n>1. The case n=2 is equivalent to the protocol of Spekkens and Rudolph with bias of 0.207. The case n=3 achieves a bias of 0.199, and n=8 achieves a bias of 0.193. The analysis of the protocols uses Kitaev's description of coin-flipping as a semidefinite program. The paper constructs an analytical solution to the dual problem which provides an upper bound on the amount that a party can cheat.13 pages, REVTeX 4 (minor corrections in v2). Arxiv paper is slightly longer than the FOCS version and contains extra examples and discussionQuantum Physicstext