Keep in mind that there are a variety of different styles and methods in making good compost. Compost bin methods, trench or sheet composting methods, lasagne gardening methods, verimcomposting, etc.
When you make compost the old classic, slow, passive, cool method, you have a wide variety of beneficial bacteria, fungi, earthworms, etc. living in them. When you make piles the fast, hot, active methods, you get more compost ready quicker, but you also sacrifice killing off some beneficial fungi and running off earthworms in the process. Cooler, passive compost methods also run the risk of letting more pathogenic or toxin substances or weed seeds, staying in the pile longer than hot, active piles. Both types of methods have their pros and cons, but it's all good, and they all work fine, based on what your individual goals and desires are in your gardening plans.
Curing is usually described as a term of letting your compost (no matter what method you choose), sit a while and build back up any beneficial fungi or earthworms, etc., to increase the microbial diversity in the pile.
You don't really have to do this perfectly in order to see great results in your gardening plans. Some gardeners just get their new freshly made compost out on their beds as a mulch, then sprinkle any form of molasses tea as a soil drench over it to get the microbes growing like crazy.
I like to use my compost made from a hot processing method after 3-4 weeks in my no-till beds as a mulch or in my aerated tea recipes. I normally use my oldest compost in my planting holes as a soil amendment.
No matter how you compost, or design your organic garden, microbes and macrobes are going to come, grow, and thrive as they digest the organic matter in and on your soil over time anyway.
The entire Kingdom of God can be totally explained as an Organic Garden (Mark 4:26)