Your description of the problem is quite a strange one - there is no reason why white spots should appear on any part of your body as a result of a sun tan. Tanning always makes the skin look darker, and at the most, if you spend a long period of time in the sun at a stretch, you may end up with a sunburn. However, this too will not cause white spots to appear on your skin.
Tanning is the result of increased melanin production by the melanocytes, the cells in our skin that produce the protective pigment we call melanin. When our skin is exposed to sunlight, more melanin is produced as a means of protection, and the skin thus becomes darker. This happens evenly over the exposed areas of skin, and never leaves white spots. A tan lasts for quite a while, but if the exposure to sunlight decreases, the tan gradually fades; once again, this too happens evenly.
Sunburn too does not result in white spots, but in inflammation of the affected areas of skin. The sunburnt skin becomes red and painful, and may later start to peel. In some more severe cases, blisters may also appear, but these are very recognizable as blisters, and are extremely unlikely to be confused for white spots.
It is therefore extremely unclear exactly what the white spots on your chest and shoulders are. It is unlikely that they are anything serious; even vitiligo (leukoderma) is quite unlikely. One very strong possibility is that they are a sign of some kind of fungal infection. These are usually nothing to worry about, but you should get your doctor to look at them, just to be sure. If they are a fungal infection, your doctor will prescribe the necessary treatment and advise you on what needs to be done.
answered by G M