Impossibility of coin flipping in generalized probabilistic theories via discretizations of semi-infinite programs

Coin flipping is a fundamental cryptographic task where spatially separated Alice and Bob wish to generate a fair coin flip over a communication channel. It is known that ideal coin flipping is impossible in both classical and quantum theory. In this work, we give a short proof that it is also impossible in generalized probabilistic theories under the generalized no-restriction hypothesis. Our proof relies crucially on a formulation of cheating strategies as semi-infinite programs, i.e., cone programs with infinitely many constraints. This introduces a formalism which may be of independent interest to the quantum community.