Stone tools for quartz or in general for agglomerate stones have some special features. Using stone tools for cutting granite is similar to cutting quartz although these should have certain features which will extend the durability of the tool. Polishing is also known to be very difficult on the dark quartz materials, you can find good solutions to this in the market, which I plan to explain in detail.
In general, we can say that the same tooling used on natural marble may be used on agglomerate marble. And for quartz agglomerate the tools are similar to the ones used on granite.
Marble based engineered stone:
Cutting: Basically any natural stone marble disc will work with agglomerate marble, and you'll notice a fine cut, as agglomerate marble is easier to work with in the shop;
Polishing edges: use the same abrasives you use on natural marble, however it may be a little more sensitive, so make sure you have good polishing abrasives, especially the yellow ones which make a great difference; In case you polish edges manually, you'll have better results with abrasives that use water, although I have seen good results with dry ones. Marble is really simple to polish, you can manually correct any defect, on the darker colours sometimes you may apply water effect chemical products to give it a nice looking final finishing.
Polishing the surface: Similar to natural marble, although you might have some difficulty in attaining the same shine, normally the shine for marble based materials is between 70-85. In some colours it may be difficult to get a perfect polish, the bigger the grain (refering to the texture of the material) used the easier it is to polish, of course darker colours are always difficult.
Quartz based engineered stone:
Cutting: quartz is much harder than granite, so expect it to consume a bit more diamond. A good diamond disc for cutting quartz should have a special diamond feature on the side of the disc as my picture below (marked in red).
Polishing edges: Try the diamond flexible pads on you quartz edges, it has given me the best results, they are expensive but they'll give you a very good finish. Many salesmen will show you dozens of solutions, however they are trying to sell the products they carry, I have tested normal abrasives, cyclone type, moving heads on edge polishers, and the ones that work best as I mentioned are the flexible diamond pads. I won't talk about brands, because I have no intention on selling, but I think you should test what you may have available in your local market. You can also use the water efffect chemical products on these edges, it really improves the final finish.
Polishing the surface: Forget it, don't even try, you'll end up destroying your quartz piece.