1. Simmer blueberries, sugar, water, and rosemary in a small heavy saucepan, uncovered, stirring occasionally, until thickened and reduced, about 10 minutes.
2. Pour into a very fine sieve and save solids, if desired (makes for a great yogurt/ice cream topping). Chill syrup until cold. Divide prosecco among 6 small flutes, then pour 2 teaspoons syrup into each drink. Gently stir. Garnish with fresh rosemary and blueberries.
3. Use additional syrup for other cocktails and to flavor sparkling water or lemonade. Syrup keeps, covered and chilled, 3 days.