  We consider two weight $L^{p}\to L^{q}$-inequalities for dyadic shifts and
the dyadic square function with general exponents $1<p,q<\infty$. It is shown
that if a so-called quadratic $\mathscr{A}_{p,q}$-condition related to the
measures holds, then a family of dyadic shifts satisfies the two weight
estimate in an $\mathcal{R}$-bounded sense if and only if it satisfies the
direct- and the dual quadratic testing condition. In the case $p=q=2$ this
reduces to the result by T. Hyt\"onen, C. P\'erez, S. Treil and A. Volberg.
  The dyadic square function satisfies the two weight estimate if and only if
it satisfies the quadratic testing condition and the quadratic
$\mathscr{A}_{p,q}$-condition holds. Again in the case $p=q=2$ we recover the
result by F. Nazarov, S. Treil and A. Volberg.
  An example shows that in general the quadratic $\mathscr{A}_{p,q}$-condition
is stronger than the Muckenhoupt type $A_{p,q}$-condition.
