Separation of AM and SZK
An oracle separation is easy to show. I'm not sure which paper showed it first, but for example you can already show that there exists an oracle $X$ such that


  $\mathsf{NP}^X \not\subset \mathsf{QSZK}^X,$


which implies 


  $\mathsf{AM}^X \not\subset \mathsf{SZK}^X$


because $\mathsf{NP}$ is in $\mathsf{AM}$ and $\mathsf{SZK}$ is in $\mathsf{QSZK}$.
See arXiv:1801.08967 for one way to prove this. See the comment on page 3 of arXiv:1902.03660 for another way to show this.

Also note that in the unrelativized world $\mathsf{AM=SZK}$ would collapse the polynomial hierarchy, because $\mathsf{SZK}$ is closed under complement, and hence this implies $\mathsf{AM=coAM}$, which collapses $\mathsf{PH}$.