  We consider a family of Schr\"odinger equations with unbounded Hamiltonian
quadratic nonlinearities on a generic tori of dimension $d\geq1$. We study the
behaviour of high Sobolev norms $H^{s}$, $s\gg1$, of solutions with initial
conditions in $H^{s}$ whose
  $H^{\rho}$-Sobolev norm, $1\ll\rho\ll s$, is smaller than $\e\ll1$. We
provide a control of the $H^{s}$-norm over a time interval of order
$O(\e^{-2})$. %where $\e\ll1$ is the size of the initial condition in
$H^{\rho}$.
  Due to the lack of conserved quantities controlling high Sobolev norms, the
key ingredient of the proof is the construction of a modified energy equivalent
to the "low norm" $H^{\rho}$ (when $\rho$ is sufficiently high) over a
nontrivial time interval $O(\e^{-2})$. This is achieved by means of normal form
techniques for quasi-linear equations involving para-differential calculus. The
main difficulty is to control the possible loss of derivatives due to the small
divisors arising form three waves interactions. By performing "tame" energy
estimates we obtain upper bounds for higher Sobolev norms $H^{s}$.
