This article investigates the role of arity of second-order quantifiers in existential second-order logic, also known as 33c6ea47edb2">. We identify fragments L of 33c6ea47edb2"> where second-order quantification of relations of arity k>1 is (nontrivially) vacuous in the sense that each formula of L can be translated to a formula of (a fragment of) monadic 33c6ea47edb2">. Let polyadic Boolean modal logic with identity (c8096c91708cd4f7a8c4556305ae4" title="Click to view the MathML source">PBML=) be the logic obtained by extending standard polyadic multimodal logic with built-in identity modalities and with constructors that allow for the Boolean combination of accessibility relations. Let be the extension of c8096c91708cd4f7a8c4556305ae4" title="Click to view the MathML source">PBML= with existential prenex quantification of accessibility relations and proposition symbols. The principal result of the article is that translates into monadic 33c6ea47edb2">. As a corollary, we obtain a variety of decidability results for multimodal logic. The translation can also be seen as a step towards establishing whether every property of finite directed graphs expressible in is also expressible in monadic 33c6ea47edb2">. This question was left open in the 1999 paper of Grädel and Rosen in the 14th Annual IEEE Symposium on Logic in Computer Science.