Beta refers to the systematic risk and return.
For example, assume you are defining momentum as "stock price today > stock price 5 days" (or whatever lookback period -- which refers to the amount of data you compare the current price against). Assume you are also adding a volume filter: "volume today > volume 5 days".
If you run this report daily, you'd get a list of stocks that meet your criteria. Say you only want to pick the top 10. Now, you can test the performance of this strategy (which will give you the beta, or systematic return and risk, of the strategy). You only want to trade strategies with positive alpha, or you're just losing money. Having an edge means that you are able to improve the risk and return characteristics (e.g. your PA stats are better than your backtest).
Momentum is a profitable strategy, and the edge typically comes from 1) lookback analysis (constantly reviewing your lookback periods, for example testing the performance of 5-day, 20-day, and 60-day returns) 2) trend analysis (advanced stats, such as kalman filtering, and tying the momentum analysis to that), and 3) volume filters that are tied to positioning. Then of course, there is risk management.
Ex-post means based off data, ex-ante means based off a forecast.