- Let $S$ be a compact oriented surface. We construct homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ generalizing the constructions of Gambaudo-Ghys and Polterovich. We prove that there are infinitely many linearly independent homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ whose absolute values bound from below the topological entropy. In case when $S$ has a positive genus, the quasimorphisms we construct on $Ham(S)$ are $C^0$-continuous. We define a bi-invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric on $Ham(S)$ is unbounded.