Qualifications are based on a minimum PR requirement (or Auto Qualifier) varying by age and type of race. Three year olds and older must have a PR of 200 or higher in the current year to be eligible for Gold Cups. Marathon and SC must have a PR of 170 or higher to get into Gold Cups. Two year olds must have a PR of at least 85 or higher in the current year to get into Gold Cups/World Championship Festival. League of Legends races require a minimum PR of 200 (or Auto Qualifier win). These restrictions ensure that the very best competitors are taking the field for these legendary races.
Note: Qualifying PRs for 3yo+, SC, and marathon must occur in a G1 race. Qualifying PRs for 2yos must occur in a graded race (G3, G2, or G1).
To check your qualifications, visit your Stable and click the 'Check Qualifications' button near the bottom. This can be checked each meet or just once right before the "big" races run - that's up to you! Whenever you check it, it will run through all of your horses and mark off anyone who meets the qualification requirements.
Once qualified, a horse will show a green checkmark under the Qualified (Q) column on your stable pages. Hovering over the Q checkmark will tell you what race(s) each horse is qualified for. This tooltip will either say "Gold Cups" or "Gold Cups and League of Legends". This should make it easy to keep track of who is qualified for which races, allowing you to better plan out their campaign.
NOTE: Three year olds cannot qualify for League of Legends races, as those races are for 4yo+. If your 3yo has a 190+ PR but does not say they are qualified for League of Legends, it is not a bug.