QuickTime is much too large to "sneakily" install itself on your system. The download file is around 50MB and requires a number of user interventions to complete the install. I believe a restart is also necessary.
Visiting a site that has content requiring QT to view it will not, usually, start a download of QT.
Trying to view media that requires QT, when it is not installed, should create the same, "Unable to open" errors as you get when trying to view any file where you do not have the required program. WMP should attempt to read the .mov file.
WIthout the URL's to the offending sites it is not really possible to research your problem. Could you supply the URL's?
QT also has a decent uninstall program so getting rid of it should not be as difficult as you describe.
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Nostly I'm curious if anyone else is noticing how invasive QuickTime has become.
They now seem to have begun using embedded links in their pages; if I click on the page, and then back off from the it without even touching the links to the program or the movies, when I check the computer it turns out that QuickTime.exe has been set up, the registry is now full of bits and pieces of it, and I have to spend a ridiculous amount of time cleaning it out.
There is no warning on the pages that QuickTime is required to view, until you get to the page, and then of course it's too late.
I had a problem last year with Quicktime loading itself, taking over my WMP, and along with it came a piggy backed CoolWebSearch bit of adware. Took a month to find it, and longer to get rid of it.
Since then I've been careful to avoid the whole mess, but now and then I come across a page that is apparently directly linked to quicktime, and all you have to do is be there, and they have you.
I tried putting QuickTime in the ''forbidden'' zone, but there are hundreds of pieces of it, and just blocking some of the components still leaves the door open for the rest. There is also QT, qtt, .qt, apple (and I dont have a mac), and all the combinations as well. Does anyone have a solution for this?

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