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This post, which I saw on HN yesterday, may answer some of that for you:

https://webkit.org/blog/8048/what-spectre-and-meltdown-mean-...

Short answer: yes.

Spectre can give you read-access to the browser process via Javascript, yes, as they described in the original article: https://spectreattack.com/spectre.pdf (section 4.3 on page 6). No write access, or access to the OS internals.

As far I remember JS exploit is possible. That's why Mozilla patched Firefox already.

Not if their browser hasn’t been patched.