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For example, I'm trying to start a project on my pc, and if I'm logged in on my iCloud account on my Apple Macbook, will it update on cross-platform?
Your projects if on a local drive, will be local. If stored on a cloud service, I would then estimate they'd "update" as the local folder updates for that service.
So unless your project files are stored on the 'net on a cloud service, no.