How do I improve quickness in punching

During your workout, do short burst as fast as you can, work on double, triple strikes and increase strikes. I would add juggling, then 4 balls if possible, nunchaku. Don't forget the blocks, If you are making quick devastating blows, you may not need to worry about blocking, don't let up.
And a Speed Bag.

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Being lightning fast is a gift your born with. But you can make adjustments to increase the speed you have. Make sure you shorten up your punches. One thing I use, is bounce a racket ball in place and pop the bag and catch the ball on it's way up. (one bounce). This will help you with jabs and crosses. It's pretty hard at first. I also hit the heavy bag with a pair of sand filled ankle weights placed in my gloves. I use 3 pounds . But this will force you to shorten your punches because 3 pounds is pretty heavy . If you go much heavier than that you will lose any snap on your punches. The key is to shorten up your punches.

The speed bag can help increase repetitive punching speed. As you learn and get control of the bag, move to smaller and smaller bags. For example, You will have to punch faster to keep up with a small 8x5 size bag rather than a larger 11x8 bag.

note the change in speed bag sizes.

smaller bags are faster and will help increase repetitive punching speed.

hope this helps.