What is quality of competition?
Quality of competition is a way to measure the general strength of the players a specific player is played against. It can be represented in a variety of ways, using a plethora of different statistics to measure the strength of the opponent, but the formula for calculating it remains the same.
The most common statistic used for quality of competition metrics is relative Corsi, but it can also be used with any possession, or even time on ice, depending on what you're looking for.
Quality of competition is calculated by using the weighted average of the statistic you're using of each opponent a player faces. What does that mean? It means if a player plays 100 minutes, and 20 of them were against the same line and defense pairing, that means 20% of his quality of competition rating comes from those player's statistics.
Because of this statistic literally including every opponent a player faces, writing out a whole equation would take forever and be extremely repetitive, not to mention convoluted. All you really need to know about this statistic is that it's a weighted average of a player's opponents based on ice time.
What does quality of competition tell us?
This depends on what statistic you're using, but for relative Corsi or any possession statistic it will tell you the average strength of the player's opponents from a possession standpoint. Using time on ice, you can see how much the average opponent of a player is used.
What do these things tell us? Well, it's a combination of two things: how the player is used by their own coaching staff, and how that player is viewed by the opposing coaching staff. Essentially, you can see the average strength of a player's opponents.
What are the limitations of quality of competition?
The same limitations that apply to any statistic you're using to build a quality of competition rating also apply to the built statistic. This means that any problems that inhibit Corsi as a measurement of performance also inhibit Corsi quality of competition.
A limitation unique to quality of competition arises in coaching tactics. Not all coaches would use Corsi as a measure of player performance, and not all would use any possession statistics at all, so using those metrics to see how a coach uses a player may not actually be that effective, depending on the coach in question.
A weak possession team will naturally have very high quality of competition numbers when using possession as well, simply because they're not playing against themselves while everyone else is. This can be seen most strongly in last year's Maple Leafs statistics.
Quality of competition is a highly debated statistic not only because there isn't a ton of general agreement on what statistic to use, but also because there is serious debate about how much it affects a player's performance. This is a huge criticism because if we don't know how much if affects their performance, what's the use of it?
As always, questions and added insight in the comments section are more than welcome.