The demonstration of quantum speedup, also known as quantum computational supremacy, that is the ability of quantum computers to outperform dramatically their classical counterparts, is an important milestone in the field of quantum computing. While quantum speedup experiments are gradually escaping the regime of classical simulation, they still lack efficient verification protocols and rely on partial validation. To that end, we derive an efficient protocol for verifying with single-mode Gaussian measurements the output states of a large class of continuous variable quantum circuits demonstrating quantum speedup, including Boson Sampling experiments, with and without i.i.d. assumption, thus enabling a convincing demonstration of quantum speedup with photonic computing. Beyond the quantum speedup milestone, our results also enable the efficient and reliable certification of a large class of intractable continuous variable multi-mode quantum states.