I am looking for ideas for circuits that stay in the state (ON/OFF) even if the power gets cut off.
and should change state according to the requirement again only after the power is restored.

Basically, the state should get latched in case of power cycling.

Thanks in advance!

 

You can do that with a CMOS latch or flip-flop circuit that has a button or coin cell battery backup.
You could also use a capacitor to maintain the state if the power is off for only a short time.

Otherwise you could use a latching relay.

What's your preference?