To the trained eye, the first test would be color. Although apatite, paraiba tourmaline and blue topaz may look similar, slight nuances in their hue will give clues as to which is which. However, even experts sometimes confuse them. Try looking at their physical and optical properties closely, e.g. hardness, inclusions, pleochroism etc.
Compared to Topaz (8) and Tourmaline (7-7.5), Apatite is soft. It is only 5 on the MOHs scale of hardness. However, as testing for hardness is destructive and it probably isn't practical to scratch your stones, look for other signs. Are the facet edges worn or adraded? Any chips, fractures? Does the surface look like it needs re-polishing?
Tourmaline can be a tad brittle, and may show some signs of wear as well. However, Paraiba varities tend to be more included than both Apatite and Topaz. Typical inclusions: channel-like trichites, needles and crystals. Also, rotate the stone to see if there are directional differences in color (pleocroism). Tourmaline has esecially high pleochroism. Paraiba Tourmaline also ranges in shades of neon green to neon blue. Blue is of course more valuable.
Unlike apatite and paraibe tourmaline, blue topaz is almost always eye clean. It is also not unusual to find them in large sizes. You may see cleavage in Topaz. Look for what looks like a straight plane in the stone. If the stone has been cut with the table perpendicular to its main crystal axis (c-axis), the cleavage plane should be parallel to the table.
If you have a photo, let me know and I can try to identify them for you. If not, I hope my answer helps!
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