Karenfrompark, lets check a couple of things. Right click on My Computer then click on the General tab. Where it shows - System: What service pack of Windows is installed? You need service pack 2 or later for XP.
Also go to C:\Program Files right click on QuickTime and go to Properties. On size on disk: What size is showing? It should show as 73.5 MB
It could be caused by a Firewall. Disable your internet security and then see if QT will open.
Are you an administrator account? It could be a permissions issue. Open the User Accounts control panel to see what type of account you have. Open the QT control panel to File Types. Does it show programmatically set? If so that will indicate permissions have been set by an administator.
Also are there too many applications running at once or loading at start up? Look down at the bottom right corner of the screen, is it loaded with icons? If so close them all out and stop them from running at start up, then see if QT will open. The only application that should load at start up is your internet security and maybe volume control, anything else you can open as needed.
If none of the above is a solution than I can help you uninstall QT and install, if so what version of QT is on the computer now. Open the QT control panel and click on About QuickTime