What is the roll of the Queen of England in Canada and other Commonwealth nations?

I know that her roll is ceremonial, but is she the official head of state in ceremony if not practice?
Does she swear in the new Canadian Prime Minister?
I thought I saw that the political leader in the hosting country of the Olympics breaks the champaign bottle to start the olympics or something. Does the Queen do that for Canada? Anybody know? What else does the Queen do? She is on Canadian coins, right? Anything else?
Thanks

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