I am having the same problem as you.* New install on three different computers (2 iMac's and Macbook Air) on Mountain Lion.* It worked well in Lion on all three, with no problems logging in or out of Mac App Store.* I have done and update from Lion, erased hard drive and started fresh install with same results.* I connected straight to the modem and through Airport Extreme, with the same results.* I am on the phone with Apple and they are unsure what is causing the problem so far.* They say it could be internet provider, but that would make no sense if it worked fine when Lion was just installed, and Lion login and out with no problems.* The tech person, John (high level tech) is being very helpful.* I sent him the error file so they can analyze the problems.* By the way, I am using Comcast(Xfinity).* I want to see of other people that are having this problem, what network they are on, to at least narrow that possibility out of the equation.
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My guess is not a lot of people who installed Mountain Lion know that this is a problem.* When first installing Mountain Lion, it ask to put in your Apple ID.* So when it starts up, the Mac App store is already logged in.* Not many people log off, unless there is multiple users.**
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So, if anyone else hasn't tried yet, log in and out of Mountain Lion Mac App Store and see if it is crashing also.