Quantum fingerprinting is a method to distinguish between two strings using quantum information. This paper shows that quantum fingerprints can be exponentially shorter than classical ones without requiring shared keys or entanglement. The scheme uses quantum states with bounded inner products to distinguish between identical and distinct fingerprints with high probability. This leads to an exponential quantum/classical gap for the equality problem in the simultaneous message passing model of communication complexity. The paper introduces a quantum protocol for the equality problem where Alice and Bob send quantum fingerprints of their inputs to a referee. The referee uses a quantum test to determine if the fingerprints are identical or distinct. The test involves a quantum circuit that measures the first qubit of a state, which allows the referee to distinguish between identical and distinct fingerprints with high probability. The paper also discusses the efficiency of quantum fingerprinting methods, showing that the number of qubits required to distinguish between 2^n quantum states with bounded inner products is logarithmic in n. This is achieved using a nonconstructive proof based on probabilistic methods. The paper further considers the state distinguishing problem, where the goal is to distinguish between two quantum states with a given inner product bound. An improved method is presented that reduces the error probability significantly compared to previous approaches. Finally, the paper considers the case where Alice and Bob share a quantum key, consisting of Bell states, and shows that this can lead to improved performance in exact fingerprinting scenarios. The results imply that quantum keys can enable errorless fingerprinting with significantly shorter fingerprints compared to classical keys. The paper concludes with acknowledgments of contributions from various researchers and references to related works in the field of quantum computing and communication complexity.Quantum fingerprinting is a method to distinguish between two strings using quantum information. This paper shows that quantum fingerprints can be exponentially shorter than classical ones without requiring shared keys or entanglement. The scheme uses quantum states with bounded inner products to distinguish between identical and distinct fingerprints with high probability. This leads to an exponential quantum/classical gap for the equality problem in the simultaneous message passing model of communication complexity. The paper introduces a quantum protocol for the equality problem where Alice and Bob send quantum fingerprints of their inputs to a referee. The referee uses a quantum test to determine if the fingerprints are identical or distinct. The test involves a quantum circuit that measures the first qubit of a state, which allows the referee to distinguish between identical and distinct fingerprints with high probability. The paper also discusses the efficiency of quantum fingerprinting methods, showing that the number of qubits required to distinguish between 2^n quantum states with bounded inner products is logarithmic in n. This is achieved using a nonconstructive proof based on probabilistic methods. The paper further considers the state distinguishing problem, where the goal is to distinguish between two quantum states with a given inner product bound. An improved method is presented that reduces the error probability significantly compared to previous approaches. Finally, the paper considers the case where Alice and Bob share a quantum key, consisting of Bell states, and shows that this can lead to improved performance in exact fingerprinting scenarios. The results imply that quantum keys can enable errorless fingerprinting with significantly shorter fingerprints compared to classical keys. The paper concludes with acknowledgments of contributions from various researchers and references to related works in the field of quantum computing and communication complexity.