This paper introduces a novel protocol for cross-platform verification of quantum states using Pauli sampling. The protocol leverages Pauli sampling, a subroutine that generates Pauli strings distributed according to their weight in the Pauli basis expansion of a quantum state. The authors show that their protocols for both Pauli sampling and cross-platform verification are efficient for quantum states with low magic and entanglement (of order O(log n)), but super-polynomial lower bounds on complexity exist for states with higher magic and entanglement. For states with real amplitudes, the protocol's requirements for cross-platform verification are significantly weakened. The paper addresses the task of distributed inner product estimation, which involves estimating the overlap between two quantum states prepared on separate devices. The authors propose algorithms for this task, demonstrating that efficient distributed estimation is possible for states with low magic and entanglement. They also show that the task remains hard even for restricted classes of input states, highlighting the importance of magic and entanglement in determining the complexity of the task. The authors introduce a Pauli sampling algorithm that uses Bell measurements on two copies of a state to generate samples from the Pauli distribution. This algorithm is efficient for states with low magic and entanglement, but requires exponentially many copies for general states. The algorithm is applied to both pure and mixed states, and the authors show that it can be used to estimate the inner product between two states. The paper also discusses the relationship between Pauli sampling and other quantum computing tasks, such as quantum learning and state tomography. The authors show that Pauli sampling can be used to estimate stabilizer entropies and to learn stabilizer states. They also compare their approach to other methods, such as matrix product state (MPS) tomography, and show that their method is more efficient in terms of resource requirements. The authors conclude that their work provides a new approach to cross-platform verification and distributed inner product estimation, with applications to a wide range of quantum computing tasks. They also highlight the importance of magic and entanglement in determining the complexity of these tasks and suggest that further research is needed to understand the full potential of Pauli sampling in quantum computing.This paper introduces a novel protocol for cross-platform verification of quantum states using Pauli sampling. The protocol leverages Pauli sampling, a subroutine that generates Pauli strings distributed according to their weight in the Pauli basis expansion of a quantum state. The authors show that their protocols for both Pauli sampling and cross-platform verification are efficient for quantum states with low magic and entanglement (of order O(log n)), but super-polynomial lower bounds on complexity exist for states with higher magic and entanglement. For states with real amplitudes, the protocol's requirements for cross-platform verification are significantly weakened. The paper addresses the task of distributed inner product estimation, which involves estimating the overlap between two quantum states prepared on separate devices. The authors propose algorithms for this task, demonstrating that efficient distributed estimation is possible for states with low magic and entanglement. They also show that the task remains hard even for restricted classes of input states, highlighting the importance of magic and entanglement in determining the complexity of the task. The authors introduce a Pauli sampling algorithm that uses Bell measurements on two copies of a state to generate samples from the Pauli distribution. This algorithm is efficient for states with low magic and entanglement, but requires exponentially many copies for general states. The algorithm is applied to both pure and mixed states, and the authors show that it can be used to estimate the inner product between two states. The paper also discusses the relationship between Pauli sampling and other quantum computing tasks, such as quantum learning and state tomography. The authors show that Pauli sampling can be used to estimate stabilizer entropies and to learn stabilizer states. They also compare their approach to other methods, such as matrix product state (MPS) tomography, and show that their method is more efficient in terms of resource requirements. The authors conclude that their work provides a new approach to cross-platform verification and distributed inner product estimation, with applications to a wide range of quantum computing tasks. They also highlight the importance of magic and entanglement in determining the complexity of these tasks and suggest that further research is needed to understand the full potential of Pauli sampling in quantum computing.