  The ancestor Gromov--Witten invariants of a compact {\Kahler} manifold $X$
can be organized in a generating function called the total ancestor potential
of $X$. In this paper, we construct Hirota Quadratic Equations (HQE shortly)
for the total ancestor potential of $\C P^1$. The idea is to adopt the
formalism developed in \cite{G1,GM} to the mirror model of $\C P^1$. We hope
that the ideas presented here can be generalized to other manifolds as well.
  As a corollary, using the twisted loop group formalism from \cite{G3}, we
obtain a new proof of the following version of the Toda conjecture: the total
descendant potential of $\C P^1$ (known also as the partition function of the
$\C P^1$ topological sigma model) is a tau-function of the Extended Toda
Hierarchy.
