Time remains one of the least well understood concepts in physics, most notably in quantum mechanics. A central goal is to find the fundamental limits of measuring time. One of the main obstacles is the fact that time is not an observable and thus has to be measured indirectly. Here we explore these questions by introducing a model of time measurements that is complete and autonomous. Specifically, our autonomous quantum clock consists of a system out of thermal equilibrium -a prerequisite for any system to function as a clock -powered by minimal resources, namely two thermal baths at different temperatures. Through a detailed analysis of this specific clock model, we find that the laws of thermodynamics dictate a trade-off between the amount of dissipated heat and the clock's performance in terms of its accuracy and resolution. Our results furthermore imply that a fundamental entropy production is associated with the operation of any autonomous quantum clock, assuming that quantum machines cannot achieve perfect efficiency at finite power. More generally, autonomous clocks provide a natural framework for the exploration of fundamental questions about time in quantum theory and beyond.

High-dimensional encoding of quantum information provides a promising method of transcending current limitations in quantum communication. One of the central challenges in the pursuit of such an approach is the certification of high-dimensional entanglement. In particular, it is desirable to do so without resorting to inefficient full state tomography. Here, we show how carefully constructed measurements in two bases (one of which is not orthonormal) can be used to faithfully and efficiently certify bipartite high-dimensional states and their entanglement for any physical platform. To showcase the practicality of this approach under realistic conditions, we put it to the test for photons entangled in their orbital angular momentum. In our experimental setup, we are able to verify 9dimensional entanglement for a pair of photons on a 11-dimensional subspace each, at present the highest amount certified without any assumptions on the state.Quantum communication offers advantages such as enhanced security in quantum key distribution (QKD) protocols [1] and increased channel capacities [2] with respect to classical means of communication. All of these improvements, ranging from early proposals [3] to recent exciting developments such as fully device-independent QKD [4, 5], rely on one fundamental phenomenon: quantum entanglement. Currently, the workhorse of most implementations is entanglement between qubits, i.e., between two-dimensional quantum systems (e.g., photon polarization). However, it has long been known that higher-dimensional entanglement can be useful in overcoming the limitations of qubit entanglement [6, 7], offering better key rates [8], higher noise resistance [9, 10] and improved security against different attacks [11].Attempting to capitalize on this insight, recent experiments have successfully generated and certified highdimensional entanglement in different degrees of freedom. In particular, the canonical way of generating two-dimensional polarization entanglement in downconversion processes already offers the potential for exploring entanglement in higher dimensions. This can be achieved by exploiting spatial degrees of freedom [12, 13], orbital angular momentum (OAM) [14-16], energy-time based encodings [17][18][19][20], or combinations thereof in hyper-entangled quantum systems [21,22]. High-dimensional quantum systems have recently also been explored in matter-based systems such as Cesium atoms [23] and superconducting circuits [24]. Thus, high- dimensional quantum systems are not only of fundamental interest but are also becoming more readily available.In this context, the certification and quantification of entanglement in many dimensions is a crucial challenge since full state tomography (FST) for bipartite systems of local dimension d requires measurements in (d + 1) 2 global product bases (i.e., tensor product bases for the global state) [25], which quickly becomes impractical in high dimensions. Due to the complexity of realizing measurements in high-dimensional spaces, previous ex...