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We study the (quantum) security of pseudorandom generators (PRGs) constructed from random oracles. We prove a "lifting theorem" showing, roughly, that if such a PRG is unconditionally secure against classical adversaries making polynomially…

The log-rank conjecture is a longstanding open problem with multiple equivalent formulations in complexity theory and mathematics. In its linear-algebraic form, it asserts that the rank and partitioning number of a Boolean matrix are…

A line of work initiated by Terhal and DiVincenzo and Bremner, Jozsa, and Shepherd, shows that quantum computers can efficiently sample from probability distributions that cannot be exactly sampled efficiently on a classical computer,…

Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a…

The Casas-Alvero conjecture predicts that every univariate polynomial over an algebraically closed field of characteristic zero sharing a common factor with each of its Hasse-Schmidt derivatives is a power of a linear polynomial. The…

We present a symbolic perturbation scheme for arbitrary polynomial geometric predicates which combines the benefits of Emiris and Canny's simple randomized linear perturbation scheme with Yap's multiple infinitesimal scheme for general…

We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework…

What does it take for real-deterministic c-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic c-valued variables out of the weak values…

In this paper, we investigate the solubility of homogeneous polynomial equations. The work of Browning, Le boudec, Sawin [3] shows that almost all homogeneous equations of degree $d\geq 4$ in $d+1$ or more variables satisfy the Hasse…

Born's rule, one of the cornerstones of quantum mechanics, relates detection probabilities to the modulus square of the wave function. Single-particle interference is accordingly limited to pairs of quantum paths and higher-order…

We show that sharp thresholds for Boolean functions directly imply average-case circuit lower bounds. More formally we show that any Boolean function exhibiting a sharp enough threshold at \emph{arbitrary} critical density cannot be…

We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising…

We prove that there is a constant $C\leq 6.614$ such that every Boolean function of degree at most $d$ (as a polynomial over $\mathbb{R}$) is a $C\cdot 2^d$-junta, i.e. it depends on at most $C\cdot 2^d$ variables. This improves the $d\cdot…

Dominant areas of computer science and computation systems are intensively linked to the hypercube-related studies and interpretations. This article presents some transformations and analytics for some example algorithms and Boolean domain…

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic…

Researchers currently use a number of approaches to predict and substantiate information-computation gaps in high-dimensional statistical estimation problems. A prominent approach is to characterize the limits of restricted models of…

We prove that, to compute a Boolean function $f$ on $N$ variables with error probability $\epsilon$, any quantum black-box algorithm has to query at least $\frac{1 - 2\sqrt{\epsilon}}{2} \rho_f N = \frac{1 - 2\sqrt{\epsilon}}{2} \bar{S}_f$…

In computer science, especially when dealing with quantum computing or other non-standard models of computation, basic notions in probability theory like "a predicate" vary wildly. There seems to be one constant: the only useful example of…

We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the…

Complex scientific models where the likelihood cannot be evaluated present a challenge for statistical inference. Over the past two decades, a wide range of algorithms have been proposed for learning parameters in computationally feasible…