I am a user of Jira because I use Jetbrain's Jira (3.10 Enterprise b#260). I only use Jetbrains's IDEA product, but they have many more products that are listed in Jira.
The problem I have with Quick Search is that it shows issues across all projects, it shows issues with all statuses.
Eventually I just got fed up with it and created a filter "IDEA Search Template" which has the projects, status, and resolution fields preset with a search term "ChangeThisXYZ". So, now what I do is click on "IDEA Search Template", then click Edit filter, then change the filter text, and then click View to see the jiras.
This got me thinking that maybe the Quick Search could just allow you to set a search filter as the default settings for the Quick Search. The Quick Search would use all the settings in the filter except for the query string. The query string would still be taken from the quick search textbox.