The QuickLaunch bar  is an ASPMenu control loaded by the Masterpage.
There are two ASPmenu controls on the master page, one is the
QuickLaunch and the other is the TopNavbar. You can remove either by
using a customized masterpage that has commented out the specific
ASPmenu control that you don't want to see.

You can also remove specific sections by creating a custom site
definition and editing the NavBars section of the ONET.xml file.

But just to be clear. You are aware that the QuickLaunch bar in each of
those sites points to different local resources. For example, the
Documents section contains links that point to the Document libraries in
that particular site. When you go to another tab it points to a
different set of Document libraries. The top link bar is used for
global navigation and the Quicklaunch bar is used for more local
navigation.