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This item gives the holder a 20% chance to move first in that turn. For example, if the holder were to use Spikes and the opponent to use Taunt, then if the item were to activate, Spikes would go before Taunt irrespective of the combatants' Speed stats. However, if the opponent were to use Bullet Punch or another elevated priority attack, then Bullet Punch would go first. Similarly, a Pokemon with Prankster will move before a Pokemon using Custap Berry when using a non-attacking move. Quick Claw's activation ignores the effects of Trick Room.
Heh. Quick Claw. That one item. Nobody in their right minds would think to use it competitively, as rather like Focus Band, the chances of it activating in any meaningful situation are slim at best, and most Pokemon would prefer a Choice Scarf if they really wanted to move first. The trouble is, a lot of characters without a right mind will use it—the Battle Subway, for instance—and here, even if it only activates sometimes, when it does activate, it can be a killer. You aren't advised to use Quick Claw competitively—but if you do, be prepared for an explosion of rage from your opponent, if it does indeed activate at the right time.
Skyarrow Bridge, East Gate - Talk to Hiker.