Large language models (LLMs) are increasingly deployed on mobile devices, making energy efficiency a key deployment constraint, yet the energy impact of prompt design remains underexplored. This paper aims to understand how two prompt properties, cognitive load and phrasing pattern, shape the energy behavior of on-device LLM inference. We conduct a broad empirical study covering prompt properties, datasets, models, and devices, with phase-level profiling that separates prefill and decode energy. We find that cognitive load primarily affects the energy cost per token, while phrasing pattern affects energy largely through token usage. Our energy-quality analysis further shows that prompt design reshapes the attainable frontier differently across models, highlighting the need for model-aware prompt design in energy-efficient on-device LLM inference. Code, datasets, and scripts are available at https://amai-gsu.github.io/PromptProperty/.

Large language models (LLMs) often answer ambiguous inputs as if they had a single intended meaning. This is problematic because input ambiguity is an upstream source of uncertainty: when a prompt admits multiple valid interpretations, the key question is whether the model should answer directly at all. Existing input uncertainty methods often rely on LLM-generated statements or clarifications, but such transformations may already disambiguate the input before the ambiguity is measured. We propose a label-preserving perturbation framework for input ambiguity detection in which perturbations remain close to the original input while preserving its label. We instantiate this principle with two detectors: a black-box rule-based perturbation library and a white-box importance-guided method using model-internal signals. Experiments across four datasets in open-domain, conversational, and visual QA show that our methods outperform output-side uncertainty baselines and LLM-paraphrase-based perturbation baselines. Our results highlight the importance of detecting ambiguity before attempting to disambiguate it. Our code is available here.