Private quantum money allows a bank to mint quantum money states that it can later verify, but that no one else can forge. In classically verifiable quantum money -- introduced by Gavinsky (CCC 2012) -- the verification is done via an interactive protocol between the bank and the user, where the communication is classical, and the computational resources required of the bank are classical. In this work, we consider memoryless interactive protocols in which the minting is likewise classical, and construct a private money scheme that achieves these two notions simultaneously (i.e., classical verification and classical minting). We call such a construction a private semi-quantum money scheme, since all the requirements from the bank in terms of computation and communication are classical.

The Bitcoin payment system involves two agent types: Users that transact with the currency and pay fees and miners in charge of authorizing transactions and securing the system in return for these fees. Two of Bitcoin's challenges are (i) securing sufficient miner revenues as block rewards decrease, and (ii) alleviating the throughput limitation due to a small maximal block size cap. These issues are strongly related as increasing the maximal block size may decrease revenue due to Bitcoin's pay-your-bid approach. To decouple them, we analyze the “monopolistic auction” [8], showing: (i) its revenue does not decrease as the maximal block size increases, (ii) it is resilient to an untrusted auctioneer (the miner), and (iii) simplicity for transaction issuers (bidders), as the average gain from strategic bid shading (relative to bidding one's true maximal willingness to pay) diminishes as the number of bids increases.

A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) - the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science. The Quantum Lovasz Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. Is there an efficient way to prepare a frustration-free state under the conditions of the QLLL? Previous results showed that the answer is positive if all local terms commute. These works were based on Moser's “compression argument” which was the original analysis technique of the celebrated resampling algorithm. We generalise and simplify the “compression argument”, so that it provides a simplified version of the previous quantum results, and improves on some classical results as well. More importantly, we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is “uniformly” gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Similarly to the previous results, our algorithm has the charming feature that it uses only local measurement operations corresponding to the local Hamiltonian terms.

In this paper, we study variants of the canonical Local Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local Hamiltonian problem. The input for the Separable Local Hamiltonian problem is the same as the Local Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable Sparse Hamiltonian problem is defined similarly, but the Hamiltonian is not necessarily local, but rather sparse. We show that the Separable Sparse Hamiltonian problem is QMA(2)-Complete, while Separable Local Hamiltonian is in QMA. This should be compared to the Local Hamiltonian problem, and the Sparse Hamiltonian problem which are both QMA-Complete. To the best of our knowledge, Separable Sparse Hamiltonian is the first non-trivial problem shown to be QMA(2)-Complete.

The Lovász Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of “weakly dependent” criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions. Our result immediately applies to the k - qsat problem (quantum analog of k - sat ): For instance we show that any collection of rank-1 projectors, with the property that each qubit appears in at most 2 k /( e ċ k ) of them, has a joint satisfiable state. We then apply our results to the recently studied model of random k - qsat . Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. [2009, 2010] we greatly extend the known satisfiable region for random k - qsat to a density of Ω(2 k / k 2 ). Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.

The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions.