What is the difference between Simple, Advanced and Cross Project Query?

Simple query filters artifacts against a single artifact type. They enable you to perform a quick search against an artifact type. The filters can be predefined by the project owner (by checking the 'Include custom query' option in Manage Artifact type' page) You can also use simple queries to search for artifacts containing a particular text string in any text attribute. Advanced query enables you to filter artifacts against multiple artifact types within the project. Cross project queries allow you to filter artifacts across projects and artifact types. NOTE: To be able to run a cross project query you need the right permissions in the chosen projects, and you can only query against attributes belonging to the same namespace.