Bleach tabs

"Bleach tablets" are either cal-hypo or trichlor, or some less concentrated derivative of one of them, and have the same disadvantages. Or they could be sodium percarbonate which isn't a sanitizer and wipes out chlorine.
 
Trichlor has CYA and Cal-Hypo has calcium. It's not just "in the tablets" but part of the chemical itself. The following are chemical facts independent of concentration of product or of pool size:

For every 10 ppm Free Chlorine (FC) added by Trichlor, it also increases Cyanuric Acid (CYA) by 6 ppm.
For every 10 ppm FC added by Dichlor, it also increases CYA by 9 ppm.
For every 10 ppm FC added by Cal-Hypo, it also increases Calcium Hardness (CH) by at least 7 ppm.

Some people do use Trichlor tabs if they are going away for a week if their CYA isn't too high already. If you don't have a pool cover and the chlorine demand is 2 ppm FC per day, then after one week the Trichlor tabs would add 8-1/2 ppm to CYA which may not be too bad. They are acidic, however, so you'd need to start with the pH a little higher and over that same week they would lower the Total Alkalinity (TA) by 7 ppm. So you could increase the TA somewhat before you go and have the pH start off at 7.7 or 7.8. You would then likely end up OK by the end of the week.

The reason there are no tablets that don't add either CYA or CH is that the chemistry does not allow for it. There is a powder/granular form of chlorine that doesn't add CYA or CH called lithium hypochlorite, but it is very expensive and doesn't form tablets (it dissolves too readily). Even Cal-Hypo tabs tend to dissolve too quickly and have extra "binders" to try and prevent that, but tend to leave residues as a result.
 
For use in food contact surfaces the concentrations are far higher than in pool use and their contact times are long so they can be considered to be bactericidal. This is not the case in pools where their concentration is too low and their kill times too long (i.e. they do not pass EPA DIS/TSS-12) though as James points out for pools they are registered as algaecides. Such "linear quats" tend to foam which is why we recommend Polyquat instead if you're going to use that kind of product. Polyquat also has clarifying properties. Neither replaces an EPA-approved disinfectant such as chlorine, however.
 
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