#paper # Abstract For every $1\leq n<\omega$ and cardinal $\kappa$ let $L^{n}_{\kappa}=L_{\omega \omega}(Q^{n}_{\kappa})$ be the logic obtained by adding the [[Magidor-Malitz quantifiers|Magidor-Malitz quantifier]] $Q^{n}_{\kappa}$. **Theorem** For every cardinal $\kappa>\omega$ and every $n<\omega$, $L_{\kappa}^{n}<L_{\kappa}^{n+1}$. I.e. there are models satisfying the same sentences of $L_{\kappa}^{n}$ but not of $L_{\kappa}^{n+1}$. **Proof idea.** The proof is by explicitely constructing models $\mathfrak{A},\mathfrak{B}$ in a language with a single $n+1$-ary predicate $R$. The constructions are done inductively, essentially ensuring that if some tuple satisfies $R$ then the elements cannot come from the same level in the construction. $\mathfrak{A}$ is constructed in $\kappa$ many steps while $\mathfrak{B}$ in $\omega$ many steps, which will ensure that $\mathfrak{A}\vDash Q^{n+1}_{\kappa}x_{0},\dots,x_{n}R(x_{0},\dots,x_{n})$ while $\mathfrak{B}\vDash \neg Q^{n+1}_{\kappa}x_{0},\dots,x_{n}R(x_{0},\dots,x_{n})$. The hard part is showing that they are $L_{\kappa}^{n}$-equivalent.