Mash blackberries in a saucepan with a potato masher. Stir in sugar until juices form; place about 1 tablespoon blackberry juice in a small bowl and stir in cornstarch. Pour cornstarch mixture into saucepan.
Bring berries to a boil, stirring often, until jam is thickened, about 15 minutes. Stir in cinnamon and allspice. Remove from heat and allow to cool. Transfer jam to a bowl, cover, and refrigerate until chilled. Stir in lemon juice.