  We extend the wellposedness results for second order backward stochastic
differential equations introduced by Soner, Touzi and Zhang \cite{stz} to the
case of a bounded terminal condition and a generator with quadratic growth in
the $z$ variable. More precisely, we obtain uniqueness through a representation
of the solution inspired by stochastic control theory, and we obtain two
existence results using two different methods. In particular, we obtain the
existence of the simplest purely quadratic 2BSDEs through the classical
exponential change, which allows us to introduce a quasi-sure version of the
entropic risk measure. As an application, we also study robust risk-sensitive
control problems. Finally, we prove a Feynman-Kac formula and a probabilistic
representation for fully nonlinear PDEs in this setting.
