You can't really turn it on from a cold boot with your phone - not easily anyway. You can do Wake on Lan with 'magic packets' but that really only works if the machine is in a sleep or hibernate state.
from a cold start? You need what's called Out-Of-Band management. it's a common feature on most servers, i.e. HP iLO, Dell DRAC etc. however the OOBM card requires constant power and network, as well as physical connection to your computer.
Turning on your pc is still an electro-mechanical process, when you push that physical power switch, it effectively shorts 2 pins in the PSU connector (look up the paperclip test) which then provides the power for the rest of the system to boot up. Unless you want to create some bizarre device that you can operate remotely to physically push the switch for you - OOBM is your best bet.