  We prove that either the images of the mapping class groups by quantum
representations are not isomorphic to higher rank lattices or else the kernels
have a large number of normal generators. Further we show that the images of
the mapping class groups have nontrivial 2-cohomology, at least for small
levels. For this purpose we considered a series of quasi-homomorphisms on
mapping class groups extending previous work of Barge and Ghys and of Gambaudo
and Ghys. These quasi-homomorphisms are pull-backs of the
Dupont-Guichardet-Wigner quasi-homomorphisms on pseudo-unitary groups along
quantum representations.
