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There is not much difference between quick reply and reply. Quote is when you want to respond to someone's post. You press on quote and you get the required post for which you want to respond. It is better you click on each one of them and try it out yourself..
I think reply and quick reply is same. Quote is to be used when you want to respond..agree, disagree or want to negate or support something from the post. You press quote and that copies the entire quote to which you want to respond. It is like screenshot in WhatsApp.