  We investigate weak coin flipping, a fundamental cryptographic primitive
where two distrustful parties need to remotely establish a shared random bit. A
cheating player can try to bias the output bit towards a preferred value. For
weak coin flipping the players have known opposite preferred values. A weak
coin-flipping protocol has a bias $\epsilon$ if neither player can force the
outcome towards their preferred value with probability more than
$\frac{1}{2}+\epsilon$. While it is known that all classical protocols have
$\epsilon=\frac{1}{2}$, Mochon showed in 2007 [arXiv:0711.4114] that quantumly
weak coin flipping can be achieved with arbitrarily small bias (near perfect)
but the best known explicit protocol has bias $1/6$ (also due to Mochon, 2005
[Phys. Rev. A 72, 022341]). We propose a framework to construct new explicit
protocols achieving biases below $1/6$. In particular, we construct explicit
unitaries for protocols with bias approaching $1/10$. To go below, we introduce
what we call the Elliptic Monotone Align (EMA) algorithm which, together with
the framework, allows us to numerically construct protocols with arbitrarily
small biases.
