Yesterday my Windows PC was starting up slowly, and when I looked in the task list I saw an entry at the top for GiantAntiSpywareUpdater and then it went away. I have never heard of GiantAntiSpyware and do not remember ever installing it. Is this some Trojan program or something that is supposed to be there?

It sounds like you have the Microsoft AntiSpyware program installed on your system. Microsoft bought and re-branded the GiantAntiSpyware program as its own. You can read the acquisition announcement.

As you noticed, there are still parts of the software where you can see Giant in the file names. The important thing to note is that what you saw is part of your anti-spyware protection and not a threat.

Those using the paid subscription service from Giant are advised to continue using that software for the duration of their subscription, because there are platform, support and functionality features in GiantAntiSpyware that are not included in the Microsoft Windows version.

In addition to running the Microsoft AntiSpyware program, you should stay current with your anti-virus updates and consider occasionally running another spyware cleaner, such as Ad-Aware or Spybot Search & Destroy. While Microsoft's AntiSpyware program is based on a product with a good reputation, not every cleaner program can catch every kind of spyware.