In Miui ROMs, there are four options for rebooting a phone from the power menu:
What does "Hot Boot" mean, technically speaking? Are there any circumstances where a hot boot could cause harm? Are there any reasons for choosing to hot boot at all?
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If it works like other "hot boot" apps and functions, then it basically restarts the Android GUI processes/threads. There was an XDA news post a while back after someone made an app for it where the app author described it as such:
Hot reboot by rebooting only the graphical Android shell.
From what I know it's sort of equivalent to restarting your X server in a Linux environment (like hitting CTRL + ALT + BACKSPACE, for example). I would definitely doubt that you'd cause any damage by restarting your graphical environment, especially considering other OSes do this pretty easily, but I suppose there's an outside chance if MIUI implements it strangely.
Typically the reason people use a "hot boot" option is if the UI has somehow gotten sluggish and would benefit from a restart but they want to save time by not restarting the whole system (maybe a memory leak in the UI or something of that sort). It could also be useful if you need to restart the UI to apply some sort of change, like a theme perhaps.