Okay so I'd like to ask what the purpose of quarter blankets are? I see everyone using them and I don't know whether I should use one or not.

They help the horse warm up when it's cold out (though you take it off after warm up) and you can use it over your legs to keep you warm
Where we are it's really cold so sometimes we use them for the whole ride. On warmer days ~0 degrees It's really important to take them off after warming up so that the horses don't become to hot, and sweat for no reason, making them colder.
Quarter sheets are nice because they allow the horses back, and hind quarters to warm up faster, meaning they don't stiffen up or stay tight due to the cold. They allow their muscles to become more supple and loose faster.
to help the horse warm up or stay warm. however there are also often used as more of a fashion statement rather than just for the real purpose
Quater sheets are used to help the horse stay warm while they get warmed up depending on how cold it is you can keep the quater sheet on for your whole ride or just the waem up