In a large saucepan, combine 10 cups of water, the sugar, and ginger and bring to a boil. Simmer for 5 minutes, remove from heat, and add the persimmons. Let sit for at least 2 hours to allow the flavors to steep. Strain into a decanter. Cover and refrigerate up to 3 days ahead. Serve chilled in teacups garnished with pine nuts.