what quality control methods a new company should follow in order to compete with other firms that are providing the same service ?

The following are the best strategic step to take in order to compete successfully with existing firms that are providing the same service1.
Conduct market survey, study the strength and weakness of the existing products or services and improve on them.2.
Put the square peg inside the square hole.
technological know-how is very important.3.
Work and achieve nothing but the best in terms of packaging4.
Set quality control standards for your productions and make sure that the standards are well enforced.5.
Give rooms for improvement suggestions and make sure the workers are well motivated.6.
Encourage promo and adverts7.
Ensure that the quality of your products specified in your product is accurate to win customers' mind in order to have a good market share4.
Give rooms for research and improvement

I love this question.
It suggests quality control is the difference maker.
In the traditional sense, not so.
The answer regarding steps to take, that have little to do with quality control, noted earlier are closer to the base.
As a Qualit leader, i would suggest simply that a solid quality system be put in place first and foremost.
Only a very experienced quality leader should attempt such a thing.
In short, this person helps establish procedures and policies, customer voc activities, assessments, and effective measurement processes, to include kKP regimes.
While QC is very important, it isn't even your top10 consideration, in the traditional sense of the word.
If by QC you man process controls and visual management, it moves up quickly, but will not supersede QA.

Physical inspection measurements and verify if it is as per material approval, tests, approved drawings, work procedures, specifications and certifications.

there is no direct relation between quality control procedure and the competation between companies for the same product

Customer's specifications based on contract signed, in addition to standard that is applicable.