Competition prestige is based on how often the competition runs and the quality (skills) of the horses that enter into it.

The most common mistake that people make when they learn this is they immediately go and jack the difficulty of the competition super high so that only the best horses can enter. The problem here is that there are very few of those horses on the game, and since new competitions have low prestige and low kitty (prize pool/earnings), those few horses aren't entering them, which means that they sit for weeks without running, which makes the prestige drop even lower instead of going higher.

To raise your competition prestige in a sustainable way, make smaller increases in the difficulty level every month or so, and enter two of your own horses every day so that your competitions show up higher in the available listings, as the default view for competitions is by participants.

This being said, if you want to increase the prestige of your trot competition specifically, you need to determine what will work for you in terms of entering two of your own horses every day: restricting it to a specific breed or leaving it open to all breeds. French Trotters are the best trotting breed on the game, though Peruvian Pasos and Finnish also perform well. For some people, it's more feasible to leave it open to all breeds so that they don't have to breed or buy horses of a specific breed with higher skills periodically to keep up with the increasing difficulty, especially since it's something that needs to be done for all competitions to keep them running daily.

Happy to answer any further questions via PM