The celebrated “hardness vs. randomness” paradigm lets us derandomize algorithms under the assumption that certain computational problems are hard to solve. Classical applications of this paradigm led to many exciting results in complexity theory, and in recent years we have been able to overcome several barriers of the original approach, using a host of new techniques.

In this talk I will survey a small selection of recent results in hardness vs. randomness. The common theme will be a close look at reconstructive pseudorandom generators, studying the complexity of reconstruction and the possibility of deterministic reconstructions.

The talk will be high-level and will aim to assume no prior knowledge.

In distributed quantum computing, the final solution of a problem is usually achieved by catenating these partial solutions resulted from different computing nodes, but intolerable errors likely yield in this catenation process. In the first part of this talk, I would like to introduce a universal error correction scheme to reduce errors and obtain effective solutions. Then, we apply this error correction scheme to designing a distributed phase estimation algorithm that presents a basic tool for studying distributed Shor’s algorithm and distributed discrete logarithm algorithm as well as other distributed quantum algorithms (for example, distributed quantum counting algorithm and distributed HHL algorithm). In the second part, I would like to introduce a quantum computing model--new quantum pushdown automata. For defining this quantum computing model, I would present a new definition of classical pushdown automata.

We will survey several cryptographic notions of computation over encrypted data (e.g., fully homomorphic encryption), as well as their instantiations from lattices. The talk will focus on deriving a simple equation at the core of all of these constructions.

Many dualities in mathematics arise from the inherent duality of ‘syntax’ and ‘semantics’ in logic. Classical Stone duality, for example, is the syntax-semantics duality for theories of classical propositional logic, with Boolean algebras encoding the syntax of a propositional theory. The logical perspective on these ‘syntax-semantics’ dualities gives both an intuitive understanding for why mathematicians (or at least logicians) would expect these dualities to hold in the first place, as well as a framework to generalise to nearby logics.

In recent years, new ‘non-commutative’ generalisations of Stone duality have been discovered, involving inverse monoids and étale groupoids. Interestingly, this branch of duality theory was developed in the absence of a logical description. In this talk, we describe a class of logical theories whose syntax-semantics duality is given by a version of non-commutative Stone duality. Rather than originating in an exotic fragment of logic, these are theories of first-order logic which share many of the same properties as the theory of vector spaces, suggesting that non-commutative Stone duality is not so distant from classical logic as one might expect.

In this talk I will present the status on quantum communications in Portugal. From the Laboratory to the operational network what has been done through the main actors in industry, academia and public institutions in Portugal. Starting with the development of national technology under a European Defence project, DISCRETION, to the deployment of the first EuroQCI (the European Quantum Communication Infrastructure) segment in Portugal, PTQCI. I will show that quantum communications is no longer a science project, but it is on the heart of sovereignty in Europe.