Higher projective tensor products of $c_0$

Let $m,n$ be positive integers with $m<n$. Under certain assumptions on the Banach space $X$, we prove that the $n$-fold projective tensor product of $X$, ${\widehat{\otimes }}_{\pi }^{n}X$, is not isomorphic to any subspace of any quotient of the $m$-fold projective tensor product, ${\widehat{\otimes }}_{\pi }^{m}X$. In particular, we prove that ${\widehat{\otimes }}_{\pi }^n{c}_0$ is not isomorphic to any subspace of any quotient of ${\widehat{\otimes }}_{\pi }^m{c}_0$.