Why are prices published without tax in the US?

I would say that because the law is not on the consumer side in the USA and therefore does not require the total price to be displayed.

Most shops will therefore leave off taxes etc as you are then more likely to buy an item. (Trustworthily companies loose trade due to other companies misleading consumers on prices, so therefore quickly all the companies become as bad as each other.)

One of the reasons this is done by sellers is so that consumers know who to blame for the prices they are paying. In particular, the seller wants the consumer to know that it is not the seller's fault that the product costs 10% more than it needs to (or whatever the rate is). Thus, the seller lists how much of the total price is attributable to sales tax so that the consumer knows that at least that portion of the price is attributable to the government. This is explained in greater detail below.

Sales taxes (and VATs) are levied on sellers, not on consumers. However, if the seller is going to have to pay a tax for selling you the product, they are going to have to charge more for the product in order to offset this additional cost to them. Specifically, if the tax costs the seller $X, then the seller is going to have to increase the price of the product by $X in order to break even. However, recognizing that consumers are price-sensitive and that they will not be happy about paying $X more for the product, the seller wishes to direct this consumer angst away from themselves. In essence, the seller is trying to protect themselves from the backlash resulting from the higher price by pointing out that they are not the reason for the higher price, the government is the reason.

In furtherance of the goal of directing consumer angst away from themselves and toward the government, the seller advertises the "pre-tax" price. This communicates to the consumer how much the seller would have been willing to sell the product for if only they didn't have to pay that darned tax.

Of course, the above-explains why the seller wants to list separately the pre-tax price and the amount attributable to tax, but it doesn't explain why the seller does not also list the post-tax price in addition to the other information. This, I assume, is done because the sellers believe that listing the higher price will have a negative effect on buying decisions. Sellers believe that consumers are affected emotionally by the listed prices, even when the consumer knows logically what the ultimate price will be. In particular, sellers believe that, even if a consumer logically knows that the ultimate price will be $Y, they will be more emotionally inclined to purchase the product if what they see on the label is less than $Y. The theory is that at least part of the emotional reaction of the consumer is tied to the price as seen, even when the consumer knows that price will increase at checkout. This is the same reason sellers prefer the ".99" format (i.e., $9.99 is preferred to $10.00), because they believe a buyer who sees $9.99 will be more emotionally receptive than one who sees $10.00, even though logically the prices are essentially the same. Thus, if listing the total (tax included) price makes consumers less emotionally receptive to the product, then sellers will be inclined to not list the total price if they can avoid it.

In addition, listing three pricing information items (base price, sales tax, total price) next to each product might not be desirable (or even feasible) in certain settings, such as limited space settings (e.g., menu board of a fast-food restaraunt). If there is only room for one priceing infomration item, then the seller is obviously going to favor the lower number.

First of all, there's hardly anywhere in the United States with a sales tax over 10% much less a rate of over 50%; you either misunderstood the price of your apples or were overcharged.

Retail sales tax in the US and Canada is inherently different than VAT (value added tax) charged many other places, sales tax is charged based on what the retail sale of a product vs. VAT accumulating through each stage of the production process.

While now easily surmountable, before computerization various local (city and county) sales taxes across the country posed a burden to retailers who operate in multiple places to calculate prices. Another reason for tax being is giving transparency as to how much tax governments are collecting.

As pointed out by others, its the fact that consumer protections are weaker in the USA, and since consumers are now used to it, and do not realise that they are probably spending more than they probably would otherwise with full price, there is very little incentive to change.

The retailers have no incentive to change, since the complaints are from a minority of consumers, the government has also no incentive as full price would perhaps impact consumption negatively at first, and consumers do not have a loud voice on this as they are mostly accustomed to it.

In most if not all states, the sales tax is levied on the total value* of the purchase, not on the individual items. For example, RCW 82.08.020:

(1) There is levied and collected a tax equal to six and five-tenths percent of the selling price on each retail sale in this state

Consider a purchase of a dozen $1 widgets with a sales tax of 8.6%: if the tax was per-item rather than per-purchase, you'd be paying $0.09 tax per item ($0.086, rounded to the nearest cent), for a total of $1.08. Computing the tax on the purchase instead gives you $12 * 0.086 = $1.03 in tax.

* Total taxable value. Food, for example, is frequently not subject to sales tax, so if you purchased bread, eggs, milk, napkins, and paper towels, the tax would only be computed for the total price of the napkins and paper towels.